unit 1 espressions, properties, linear, inequalities · 2018-10-17 · unit 1 espressions,...
TRANSCRIPT
Unit 1 Espressions properties linear inequalities
Topic 1
Expressions and Properties
An Algebraic Expression consists of ______________________
______________________ and __________________________
A _________________ is a symbol used to represent an unidentified number or value
A ___________ may be a number variable or a product or quotient of numbers and variables
Unit 1 Espressions properties linear inequalities
Examples
Nonshyexamples
Writing Expressions
Addition
Subtraction
Multiplication
Division
Unit 1 Espressions properties linear inequalities
Examples
3x4
5z2 + 16
16 times u to the second power minus 3
one half a plus the quotient of 6 times b and 7
You Try
a) A number t more than 6 b) 10 less than the product of 7 and f
c) Two thirds of the volume v d) a2 shy 18b
e) 2a + 6 f) w shy 24v
Unit 1 Espressions properties linear inequalities
Bell RingerPut HW in Basket
Write the Algebraic expression for
twice the difference of p and q
Write the phrase for
Evaluating Expressions
Order of Operations
P
E
M
D
A
S
Examples
a) 16 shy 8 divide 22 + 14
b) 15 shy [10 + (3 shy 2)3] + 6
c) 2x + 5 when x = 3
Unit 1 Espressions properties linear inequalities
You try
a) 5 4(10 shy 8)2 + 20 b) 3h + 2(4 shy p)3 when h = 2 and p = 1
c) t2(3r + 5) divide s where r = 6 s = 4 and t = 2
Bell RingerHomework on Desk
Evaluate 3(x + 4)2 shy 6y divide 3 When x = 1 and y = 7
Unit 1 Espressions properties linear inequalities
Properties
What properties do you remember
Reflexive Any quantity is equal to itself a = a4 +7 = 4 + 7
Symmetric If a = b then b = aIf 8 = 2 + 6 then 2 + 6 = 8
Transitive If a = b and b = c then a = cIf 6 + 9 = 12 + 3 and 12 + 3 = 15 then 6 + 9 = 15
Substitution If a = b then a may be replaced by b If n = 11 then 4n = 4 11
Additive Identity
Additive Inverse
Multiplicative Identity
Multiplicative Inverse
Multiplicative Propertyof Zero
Unit 1 Espressions properties linear inequalities
Commutative Property
Associative Property
Distributive Property
Name the Property
a) (5 1) = 5
b) 6 + (shy6) = 0
c) 5 + 7 = 7 + 5
d) 3(2b) = (32)b
e) 4(3 + x) = 12 + 4x
Unit 1 Espressions properties linear inequalities
More Distributive Property
a) 7 49 = 7(50 shy 1) b) 304(15) = (300 + 4)(15)
c) (8 + 4n)2 d) shy6(r + 3g shy t)
e) 4k + 8k = (4 + 8) k f) 6a2 + a2 + 2a
Bell RingerHW out on your Desk
Write an example of the Multiplicative Inverse property and the Distributive Property
Unit 1 Espressions properties linear inequalities
Bell Ringer
1) Simplify 2(3x shy 4y)
2) Write an example of the transitive property
Bell Ringer
Check if the equation y = 3x + 1 is true for the following values of x and y
a) x = 0 y = 1
b) x = shy2 y = 5
c) (3 10)
Unit 1 Espressions properties linear inequalities
Topic 2
Functions
The coordinate system is formed by the intersection of two number lines the horizontal axis and the vertical axis
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Examples
Nonshyexamples
Writing Expressions
Addition
Subtraction
Multiplication
Division
Unit 1 Espressions properties linear inequalities
Examples
3x4
5z2 + 16
16 times u to the second power minus 3
one half a plus the quotient of 6 times b and 7
You Try
a) A number t more than 6 b) 10 less than the product of 7 and f
c) Two thirds of the volume v d) a2 shy 18b
e) 2a + 6 f) w shy 24v
Unit 1 Espressions properties linear inequalities
Bell RingerPut HW in Basket
Write the Algebraic expression for
twice the difference of p and q
Write the phrase for
Evaluating Expressions
Order of Operations
P
E
M
D
A
S
Examples
a) 16 shy 8 divide 22 + 14
b) 15 shy [10 + (3 shy 2)3] + 6
c) 2x + 5 when x = 3
Unit 1 Espressions properties linear inequalities
You try
a) 5 4(10 shy 8)2 + 20 b) 3h + 2(4 shy p)3 when h = 2 and p = 1
c) t2(3r + 5) divide s where r = 6 s = 4 and t = 2
Bell RingerHomework on Desk
Evaluate 3(x + 4)2 shy 6y divide 3 When x = 1 and y = 7
Unit 1 Espressions properties linear inequalities
Properties
What properties do you remember
Reflexive Any quantity is equal to itself a = a4 +7 = 4 + 7
Symmetric If a = b then b = aIf 8 = 2 + 6 then 2 + 6 = 8
Transitive If a = b and b = c then a = cIf 6 + 9 = 12 + 3 and 12 + 3 = 15 then 6 + 9 = 15
Substitution If a = b then a may be replaced by b If n = 11 then 4n = 4 11
Additive Identity
Additive Inverse
Multiplicative Identity
Multiplicative Inverse
Multiplicative Propertyof Zero
Unit 1 Espressions properties linear inequalities
Commutative Property
Associative Property
Distributive Property
Name the Property
a) (5 1) = 5
b) 6 + (shy6) = 0
c) 5 + 7 = 7 + 5
d) 3(2b) = (32)b
e) 4(3 + x) = 12 + 4x
Unit 1 Espressions properties linear inequalities
More Distributive Property
a) 7 49 = 7(50 shy 1) b) 304(15) = (300 + 4)(15)
c) (8 + 4n)2 d) shy6(r + 3g shy t)
e) 4k + 8k = (4 + 8) k f) 6a2 + a2 + 2a
Bell RingerHW out on your Desk
Write an example of the Multiplicative Inverse property and the Distributive Property
Unit 1 Espressions properties linear inequalities
Bell Ringer
1) Simplify 2(3x shy 4y)
2) Write an example of the transitive property
Bell Ringer
Check if the equation y = 3x + 1 is true for the following values of x and y
a) x = 0 y = 1
b) x = shy2 y = 5
c) (3 10)
Unit 1 Espressions properties linear inequalities
Topic 2
Functions
The coordinate system is formed by the intersection of two number lines the horizontal axis and the vertical axis
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Examples
3x4
5z2 + 16
16 times u to the second power minus 3
one half a plus the quotient of 6 times b and 7
You Try
a) A number t more than 6 b) 10 less than the product of 7 and f
c) Two thirds of the volume v d) a2 shy 18b
e) 2a + 6 f) w shy 24v
Unit 1 Espressions properties linear inequalities
Bell RingerPut HW in Basket
Write the Algebraic expression for
twice the difference of p and q
Write the phrase for
Evaluating Expressions
Order of Operations
P
E
M
D
A
S
Examples
a) 16 shy 8 divide 22 + 14
b) 15 shy [10 + (3 shy 2)3] + 6
c) 2x + 5 when x = 3
Unit 1 Espressions properties linear inequalities
You try
a) 5 4(10 shy 8)2 + 20 b) 3h + 2(4 shy p)3 when h = 2 and p = 1
c) t2(3r + 5) divide s where r = 6 s = 4 and t = 2
Bell RingerHomework on Desk
Evaluate 3(x + 4)2 shy 6y divide 3 When x = 1 and y = 7
Unit 1 Espressions properties linear inequalities
Properties
What properties do you remember
Reflexive Any quantity is equal to itself a = a4 +7 = 4 + 7
Symmetric If a = b then b = aIf 8 = 2 + 6 then 2 + 6 = 8
Transitive If a = b and b = c then a = cIf 6 + 9 = 12 + 3 and 12 + 3 = 15 then 6 + 9 = 15
Substitution If a = b then a may be replaced by b If n = 11 then 4n = 4 11
Additive Identity
Additive Inverse
Multiplicative Identity
Multiplicative Inverse
Multiplicative Propertyof Zero
Unit 1 Espressions properties linear inequalities
Commutative Property
Associative Property
Distributive Property
Name the Property
a) (5 1) = 5
b) 6 + (shy6) = 0
c) 5 + 7 = 7 + 5
d) 3(2b) = (32)b
e) 4(3 + x) = 12 + 4x
Unit 1 Espressions properties linear inequalities
More Distributive Property
a) 7 49 = 7(50 shy 1) b) 304(15) = (300 + 4)(15)
c) (8 + 4n)2 d) shy6(r + 3g shy t)
e) 4k + 8k = (4 + 8) k f) 6a2 + a2 + 2a
Bell RingerHW out on your Desk
Write an example of the Multiplicative Inverse property and the Distributive Property
Unit 1 Espressions properties linear inequalities
Bell Ringer
1) Simplify 2(3x shy 4y)
2) Write an example of the transitive property
Bell Ringer
Check if the equation y = 3x + 1 is true for the following values of x and y
a) x = 0 y = 1
b) x = shy2 y = 5
c) (3 10)
Unit 1 Espressions properties linear inequalities
Topic 2
Functions
The coordinate system is formed by the intersection of two number lines the horizontal axis and the vertical axis
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell RingerPut HW in Basket
Write the Algebraic expression for
twice the difference of p and q
Write the phrase for
Evaluating Expressions
Order of Operations
P
E
M
D
A
S
Examples
a) 16 shy 8 divide 22 + 14
b) 15 shy [10 + (3 shy 2)3] + 6
c) 2x + 5 when x = 3
Unit 1 Espressions properties linear inequalities
You try
a) 5 4(10 shy 8)2 + 20 b) 3h + 2(4 shy p)3 when h = 2 and p = 1
c) t2(3r + 5) divide s where r = 6 s = 4 and t = 2
Bell RingerHomework on Desk
Evaluate 3(x + 4)2 shy 6y divide 3 When x = 1 and y = 7
Unit 1 Espressions properties linear inequalities
Properties
What properties do you remember
Reflexive Any quantity is equal to itself a = a4 +7 = 4 + 7
Symmetric If a = b then b = aIf 8 = 2 + 6 then 2 + 6 = 8
Transitive If a = b and b = c then a = cIf 6 + 9 = 12 + 3 and 12 + 3 = 15 then 6 + 9 = 15
Substitution If a = b then a may be replaced by b If n = 11 then 4n = 4 11
Additive Identity
Additive Inverse
Multiplicative Identity
Multiplicative Inverse
Multiplicative Propertyof Zero
Unit 1 Espressions properties linear inequalities
Commutative Property
Associative Property
Distributive Property
Name the Property
a) (5 1) = 5
b) 6 + (shy6) = 0
c) 5 + 7 = 7 + 5
d) 3(2b) = (32)b
e) 4(3 + x) = 12 + 4x
Unit 1 Espressions properties linear inequalities
More Distributive Property
a) 7 49 = 7(50 shy 1) b) 304(15) = (300 + 4)(15)
c) (8 + 4n)2 d) shy6(r + 3g shy t)
e) 4k + 8k = (4 + 8) k f) 6a2 + a2 + 2a
Bell RingerHW out on your Desk
Write an example of the Multiplicative Inverse property and the Distributive Property
Unit 1 Espressions properties linear inequalities
Bell Ringer
1) Simplify 2(3x shy 4y)
2) Write an example of the transitive property
Bell Ringer
Check if the equation y = 3x + 1 is true for the following values of x and y
a) x = 0 y = 1
b) x = shy2 y = 5
c) (3 10)
Unit 1 Espressions properties linear inequalities
Topic 2
Functions
The coordinate system is formed by the intersection of two number lines the horizontal axis and the vertical axis
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
You try
a) 5 4(10 shy 8)2 + 20 b) 3h + 2(4 shy p)3 when h = 2 and p = 1
c) t2(3r + 5) divide s where r = 6 s = 4 and t = 2
Bell RingerHomework on Desk
Evaluate 3(x + 4)2 shy 6y divide 3 When x = 1 and y = 7
Unit 1 Espressions properties linear inequalities
Properties
What properties do you remember
Reflexive Any quantity is equal to itself a = a4 +7 = 4 + 7
Symmetric If a = b then b = aIf 8 = 2 + 6 then 2 + 6 = 8
Transitive If a = b and b = c then a = cIf 6 + 9 = 12 + 3 and 12 + 3 = 15 then 6 + 9 = 15
Substitution If a = b then a may be replaced by b If n = 11 then 4n = 4 11
Additive Identity
Additive Inverse
Multiplicative Identity
Multiplicative Inverse
Multiplicative Propertyof Zero
Unit 1 Espressions properties linear inequalities
Commutative Property
Associative Property
Distributive Property
Name the Property
a) (5 1) = 5
b) 6 + (shy6) = 0
c) 5 + 7 = 7 + 5
d) 3(2b) = (32)b
e) 4(3 + x) = 12 + 4x
Unit 1 Espressions properties linear inequalities
More Distributive Property
a) 7 49 = 7(50 shy 1) b) 304(15) = (300 + 4)(15)
c) (8 + 4n)2 d) shy6(r + 3g shy t)
e) 4k + 8k = (4 + 8) k f) 6a2 + a2 + 2a
Bell RingerHW out on your Desk
Write an example of the Multiplicative Inverse property and the Distributive Property
Unit 1 Espressions properties linear inequalities
Bell Ringer
1) Simplify 2(3x shy 4y)
2) Write an example of the transitive property
Bell Ringer
Check if the equation y = 3x + 1 is true for the following values of x and y
a) x = 0 y = 1
b) x = shy2 y = 5
c) (3 10)
Unit 1 Espressions properties linear inequalities
Topic 2
Functions
The coordinate system is formed by the intersection of two number lines the horizontal axis and the vertical axis
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Properties
What properties do you remember
Reflexive Any quantity is equal to itself a = a4 +7 = 4 + 7
Symmetric If a = b then b = aIf 8 = 2 + 6 then 2 + 6 = 8
Transitive If a = b and b = c then a = cIf 6 + 9 = 12 + 3 and 12 + 3 = 15 then 6 + 9 = 15
Substitution If a = b then a may be replaced by b If n = 11 then 4n = 4 11
Additive Identity
Additive Inverse
Multiplicative Identity
Multiplicative Inverse
Multiplicative Propertyof Zero
Unit 1 Espressions properties linear inequalities
Commutative Property
Associative Property
Distributive Property
Name the Property
a) (5 1) = 5
b) 6 + (shy6) = 0
c) 5 + 7 = 7 + 5
d) 3(2b) = (32)b
e) 4(3 + x) = 12 + 4x
Unit 1 Espressions properties linear inequalities
More Distributive Property
a) 7 49 = 7(50 shy 1) b) 304(15) = (300 + 4)(15)
c) (8 + 4n)2 d) shy6(r + 3g shy t)
e) 4k + 8k = (4 + 8) k f) 6a2 + a2 + 2a
Bell RingerHW out on your Desk
Write an example of the Multiplicative Inverse property and the Distributive Property
Unit 1 Espressions properties linear inequalities
Bell Ringer
1) Simplify 2(3x shy 4y)
2) Write an example of the transitive property
Bell Ringer
Check if the equation y = 3x + 1 is true for the following values of x and y
a) x = 0 y = 1
b) x = shy2 y = 5
c) (3 10)
Unit 1 Espressions properties linear inequalities
Topic 2
Functions
The coordinate system is formed by the intersection of two number lines the horizontal axis and the vertical axis
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Commutative Property
Associative Property
Distributive Property
Name the Property
a) (5 1) = 5
b) 6 + (shy6) = 0
c) 5 + 7 = 7 + 5
d) 3(2b) = (32)b
e) 4(3 + x) = 12 + 4x
Unit 1 Espressions properties linear inequalities
More Distributive Property
a) 7 49 = 7(50 shy 1) b) 304(15) = (300 + 4)(15)
c) (8 + 4n)2 d) shy6(r + 3g shy t)
e) 4k + 8k = (4 + 8) k f) 6a2 + a2 + 2a
Bell RingerHW out on your Desk
Write an example of the Multiplicative Inverse property and the Distributive Property
Unit 1 Espressions properties linear inequalities
Bell Ringer
1) Simplify 2(3x shy 4y)
2) Write an example of the transitive property
Bell Ringer
Check if the equation y = 3x + 1 is true for the following values of x and y
a) x = 0 y = 1
b) x = shy2 y = 5
c) (3 10)
Unit 1 Espressions properties linear inequalities
Topic 2
Functions
The coordinate system is formed by the intersection of two number lines the horizontal axis and the vertical axis
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
More Distributive Property
a) 7 49 = 7(50 shy 1) b) 304(15) = (300 + 4)(15)
c) (8 + 4n)2 d) shy6(r + 3g shy t)
e) 4k + 8k = (4 + 8) k f) 6a2 + a2 + 2a
Bell RingerHW out on your Desk
Write an example of the Multiplicative Inverse property and the Distributive Property
Unit 1 Espressions properties linear inequalities
Bell Ringer
1) Simplify 2(3x shy 4y)
2) Write an example of the transitive property
Bell Ringer
Check if the equation y = 3x + 1 is true for the following values of x and y
a) x = 0 y = 1
b) x = shy2 y = 5
c) (3 10)
Unit 1 Espressions properties linear inequalities
Topic 2
Functions
The coordinate system is formed by the intersection of two number lines the horizontal axis and the vertical axis
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer
1) Simplify 2(3x shy 4y)
2) Write an example of the transitive property
Bell Ringer
Check if the equation y = 3x + 1 is true for the following values of x and y
a) x = 0 y = 1
b) x = shy2 y = 5
c) (3 10)
Unit 1 Espressions properties linear inequalities
Topic 2
Functions
The coordinate system is formed by the intersection of two number lines the horizontal axis and the vertical axis
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Topic 2
Functions
The coordinate system is formed by the intersection of two number lines the horizontal axis and the vertical axis
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
A point is represented on a graph using ordered pairs
bull An ordered pair is a set of numbers or coordinates written in the form ________
bull The xshyvalue called the xshycoordinate represents the _______________________ placement of the point
bull The yshyvalue called the yshycoordinate represents the ______________________ placement of the point
A set of ordered pairs is called a ___________ A __________ can be represented in several different ways
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
A mapping illustrates how each element of the domain is paired with an element in the range
The set of the _______ numbers of the ordered pairs is the ______________
The set of __________ numbers of the ordered pairs is the ______________
This mapping represents the ordered pairs (shy2 4) (shy1 4) (0 6) (1 8) amp (2 8)
Domain Range
Also called the _____________________________________
Also called the __________________
___________________
The same relation represented in many forms
Ordered Pairs
(1 2)(shy2 4)(0 shy3)
Table
x y
MappingDomain Range
Graph
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
If an ordered pair is a solution to an equation or inequality then when it is substituted into the equation or inequality the numerical sentence is true
a) Is (3 15) a solution to the equation y = 5x
b) Is (2 7) a solution to the equation 3x + 2y = 21
c) Is (shy1 4) a solution to the inequality y gt 2x shy 3
d) Is (4 shy6) a solution to the inequality 2x shy 3y lt shy4
Bell RingerDraw a mapping for the following relation
(shy2 4) (shy8 6) (6 4) (2 7) (6 shy3) (shy4 5) amp (4 shy2)
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
A Function is a relationship between input and output In a function there is exactly _____ output for each input
Every element of the _________ is paired with exactly one element of the ________
The _____ coordinates cannot repeat
Example NonshyExampleDomain Range Domain Range
Is it a functiona) Domain Range
shy2 shy3 0 6 3 4 9
c)(2 1) (3 shy2) (3 1) (2 shy2)
b)Domain 1 3 5 1
Range 4 2 4 shy4
d)
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Graphs of functions can either be discrete or continuous
Discrete functions are points that are _______________________
Continuous functions are points that are connected with a _____________ or a ______________________
You can use the __________________________________ to see if a graph represents a function
If a vertical line intersects the graph more than once then the graph is not a function
Examples of graphs
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell RingerAre the following examples of functions
a) (4 5) (3 7) (2 3) (shy1 3) (shy2 shy4) (shy3 7) (4 5) (2 5)
b)
HW Page 345
Numbers 28shy37
Bell RingerHW Due tomorrowPage 345 Numbers 28shy37 Are the following functions
b)x f(x)
a)shy2245shy7shy275
shy52539shy5136
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Function Notation
Written as f(x) = and read as f of x is
In a function x represents the element of the domain and f(x) represents the element of the range
Suppose you want to know what value in the range corresponds to the element 5 in the domain This is written f(5) and is read f of 5The value f(5) is found by substituting 5 for x in the equation
Examplef(x) = shy4x + 7a) f(2) = b) f(shy3) =
Examples
a) f(x) = 2x shy 3
f(1)
f(shy2)
6 shy f(5)
f(shy1) + f(2)
b) h(t) = shy16t2 + 68t + 2
h(4)
2[h(g)]
h(3) shy f(4)
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringera) f(x) = 2x + 4Find f(shy3)
b) g(y) = 2 shy 3yFind g(shy5)
c) Find f(g(shy3))
Bell Ringer
Get into a group of FIVE (5)
(NO you CANNOT have a group of 6 NOT 4 either FIVE)
(If you do not have a group go under the TV)
v
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer
Replace the square with the correct numerical value
1) 4 + = 15
2) 32 shy = 20
3) 15 shy = 56
Topic 3
Solving Equations
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
An equation is an algebraic sentence that contains an _________________
Expression Equation
Finding the value for a variable that makes a sentence true is called solving the sentence This replacement value is a ______________
If there is more than one element that makes the sentence true than the elements make up the ________________________
Given the set 0123 which value makes the sentence 8m shy 7 = 17 true
What is the solution set of the equation 4a = 16 from the replacement set 2 3 4 5 6
Solving an equation
You can use inverse operations in the reverse order to solve equations
4a = 16
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
When solving an equation there could be no value that makes it true one value that makes it true or every value makes it true
If there is no value that makes it true then there is no solution
If there is one value that makes it true we state that value
If every value makes it true it is called an identity and we state all real numbers
a) 3 + t = 32
b) 5(r + 2) = 5(1 + r)
c) 3(b + 1) shy 5 = 3b shy 2
Bell Ringer
Find which value(s) from the replacement set shy1 0 1 2 are a part of the solution set for shy2x + 3 lt 3 Make a table
HW on your Desk
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Solving One Step Equations
Complete inverse operations
Get the ______________ by itself
Addition Property of Equality you can add the same thing to both sides of an equation without changing the solution If a = b then a + c = b + c
a) c shy 22 = 54 b) 113 = g shy 25 c) j shy 87 = shy3
Subtraction Property of Equality you can subtract the same thing from both sides of an equation without changing the solution If a = b then a shy c = b shy c
a) 63 + m = 79 b) 27 + k = 30 c) shy12 = p + 16
MultiplicationDivision Property of Equality you can multiplydivide by the same nonzero number on both sides of an equation without changing the solution
a) 39 = shy3r b) 3k = 6 c) shy1 = 2b 5 4 3
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell RingerFill in the box with a number that completes the sentence
a) 5 shy 8 = 7
b) 2 + 3 = 29
Solving MultishyStep Equations
Solve by completing inverse operations in the reverse order
Get the __________ by itself
a) 11x shy 4 = 29 b) 5 + 4x = shy11
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
c) a + 7 = 5 d) n + 1 = 15 8 shy2
e) shy3 = 2 + a f) 3b shy 8 = 11 11 2
Consecutive integers are integers in counting order such as 4 5 6 or n n + 1 n + 2
Consecutive Integers n n + 1
Consecutive Even Integers n n + 2
Consecutive Odd Integers n n + 2
Example The sum of three consecutive odd integers is shy51
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) Find three consecutive integers with a sum of 21
b) Find four consecutive even integers with a sum of 108
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer
Among the career home run leaders for Major League Baseball Hank Aaron has 175 fewer than twice the number that Dave Winfield has Hank Aaron hit 755 home runs Write an equation for this situation How many home runs did Dave Winfield hit in his career
101514 HW on your Desk
Bell Ringer
Find the value for x that will make the perimeter of the triangle equal to the perimeter of the rectangle
101614
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Equations with Variables on both sides of the equals sign
move the smallest variable to the opposite side using addition or subtraction
Example 2 + 5k = 3k shy 6 5a + 2 = 6 shy 7a
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Homework
Due tomorrowPage 122 Numbers 13shy22
Due FridayPage 122Numbers 33shy36
Bell Ringer101714
Solve for q
24q shy 12 = 6q + 56
HW in the Basket
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bonus
1) Solve for x 3x + 2 = ax shy 4
2) Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers (Hint Let n = the lesser and n + 2 = the greater)
3) Chris has saved twice the number of quarters that Nora saved plus 6 The number of quarters Chris saved is also five times the difference of the number of quarters Nora saved and 3 Write and solve an equation to find the number of quarters they each have saved
Bell Ringer
1 Find 8 consecutive even integers with a sum of 472
2 Three times the lesser of two consecutive even integers is 6 less than six times the greater number Find the integers
(Hint Let the lesser = n and the greater = n + 2)
102114
shy
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer102414
1) Find 5 consecutive odd integers with a sum of 85
2) Solve for x 4x + 7 = 2x shy 11
HW on your desk
1) y = 4 + x
2) 2x shy 5y = 1
3) x + 2y = 4
4) shy3 + 2y = shy5
Ax + By = C
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer102714
1) Find 3 consecutive integers with a sum of shy111
2) Solve for q 3q shy 6 = 4q + 8
Topic 4
Graphing Linear Equations
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
A linear equation is an equation that forms a _______ when it is graphed
The standard form of a linear equation is Ax + By = C where C is a constant and Ax and By are variable terms
If you can write an equation in standard form then it is a linear equation if you cannot write it in standard form then it is not a linear equation
a) y = 4 shy 3x b) 6x shy xy = 4 c) y = shy1 3
The _____shyintercept is where the graph crosses the yshyaxis
The _____shyintercept is where the graph crosses the xshyaxis
When graphing a linear equation you must always have
a) a straight line
b) arrows (if no restriction is present)
c) label
d) table
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) Find the xshyintercept by letting y = 0
2) Find the yshyintercept by letting x = 0
3) Plot the two points and connect them
2x + 4y = 16
a) shyx + 2y = 3
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
103114
Login shy Your bell ringer will be sent to you
Bell Ringer103014
Get the following in standard form
a) 3y + 4 = 2x shy 3
b) 2x shy 3y + 5 = 3x shy y shy 3
HW on Desk
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
1) 2x + 4y = 8
2) 3y = 6x + 12
3) 2y shy 4 = 3x + 2
4) 5y + 3x shy 7 = 6y + x + 1
Bell Ringer11214
Login your bell ringer will be sent to youShow your work on the bell ringer sheet
Place any old HW in the basket
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Rate of Change
Change in yChange in x
a)Number of Computer Games
(x)
Total Cost (y)
2 78
4 156
6 234
Number Floor Tiles
(x)
Area of Tiled Surface (y)
3 48
6 96
9 144
b)
If the rate of change is constant then the table represents a linear function
a)(x) (y)
1 shy6
4 shy8
7 shy10
10 shy12
13 shy14
(x) (y)
shy3 10
shy1 12
shy1 12
1 16
3 18
b)
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
The ________ of a nonvertical line is the ratio of the change in the yshycoordinate (rise) to the change in the xshycoordinate (run) as you move from left to right
Slope = Rise or Δy or change in yshycoordinates Run Δx change in xshycoordinates
Slope Formula a) (shy1 3) and (2 shy2)
m = y2 shy y1x2 shy x1
b) (shy2 0) and (15) c) (shy3 4) and (2 shy3)
d) (shy3 shy1) and (2 shy1) e) (3 6) and (4 8)
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
HW
Page 351 numbers 51 and 52 32shy34
Page 362 numbers 9shy11
Bell Ringer102814
Find the slope of the lines that go through the following points
1) (shy2 4) amp (5 7)
2) (3 6) amp (shy1 9)
Find the missing yshyvaluem = 1 (2 7) amp (6 y)
2
HW on Desk
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Positive Slope Negative Slope
Slope of Zero Undefined Slope
SlopeshyIntercept Form
y = mx + b Where m is the slope and b is the yshyintercept
a) 3x + 2y = 6 b) 3x shy 4y = shy12
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
To graph a linear equation
1) solve the equation for y (get it into slopeshyintercept form)
2) graph the yshyintercept
3) use the slope to rise and run
4) label
a) shy2x + 5y = 10
a) shy4x + y = 2 b) 2x + y = shy6
c) shy3x + 7y = 21 d) 3y = 15
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
HW
Page 351 Numbers 32shy34 and 36shy38
Bell Ringer103014
Get the following in slopeshyintercept form
a) 3x shy 2y shy 2= x + 14
b) 5 + 3x + 4y = 6 shy 7x
HW on Desk
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
Write the following equation in slopeshyintercept form
3x + 2y = 9 shy 2x
Graph it on your calculator and copy down 4 points from the table
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Point Slope Form
y shy y1 = m(x shy x1)
where m is the slope x1 and y1 come from the point and x and y are variables
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line going through the point (3 shy2) with a slope of 2
b) Write an equation of the line going through the point (1 shy3) with a slope of shy4
Graphing using pointshyslope form
1) get an equation for the line in pointshyslope form
2) graph the given point
3) use the slope to rise and run to the next points
4) label
a) Graph the line that has a slope of 5 and passes through the point (6 shy2)
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer11414
1) Write an equation in pointshyslope form of the line going through the point (4 shy5) and has a slope of shy2
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Writing Equations
Given the slope of a line and a point on the line use the pointshyslope form
If required solve into slopeshyintercept or standard form
Given two points on the line find the slope using the slope formula then use the pointshyslope form with one of the given points and the slope you found
If required solve into slopeshyintercept or standard form
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) Write an equation of the line that passes through (21) with a slope of 3
b) Write an equation of the line that passes through (shy2 5) with a slope of 3
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
c) Write an equation of a line that passes through (shy4 7) with a slope of shy1 in slopeshyintercept form
2 points
a) Write an equation of the line that passes through (31) and (24) in standard form
b) Write an equation of the line that passes through (shy4shy2) and (shy5shy6) in slopeshyintercept form
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
c) Write an equation of the line that passes through (shy1 12) and (4 shy8)
d) Write an equation of the line that passes through (5 shy8) and (shy7 0)
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer11514
1) Write an equation of the line passing through the point (shy3 shy6) with a slope of shy2 in standard form
2) Write an equation of the line passing through the point (shy2 shy4) with a slope of 3 in slopeshyintercept form
Writing equations only using slopeshyintercept form
Given 1 point and the slopeStart with the formulaReplace m x and y with your given valuesSolve for bWrite formula with given m and your found b
Given 2 pointsUse the slope formula to find mSolve like you would if you were given 1 point and the slope
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) Write an equation of a line that passes through (2 shy3) with a slope of 1
2
b) Write an equation of the line that passes through (shy3 shy4) and (shy2 shy8)
c) Write an equation of the line that passes through (6 shy2) and (3 4)
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer11714
Write an equation of the line going through the points (shy5 2) and (shy5 7)
Write an equation of a line going through the points (shy6 8) and (shy6 8)
Parallel Lines Lines in the same plane that do not intersect
Parallel lines have the same ______________
a) Write an equation in slope intercept form for the line that passes through (shy3 5) and is parallel to the graph of y = 2x shy 4
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
b) Write an equation of the line passing through the point (shy3 4) and is parallel to the xshyaxis
c) Write an equation of the line passing through the point (4 shy2) and is parallel to the graph y = 3x shy 6
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Hw in the basket
Take out a piece of graph paper and create four coordinate grids on it
Write and graph an inequality to represent each situation
1) Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14 while a set of bed linens costs $60 How many gallons of paint and bed linens can she buy
2) The soccer team is selling hot dogs for a dollar and hamburgers for a dollar fifty to raise money to purchase new goals which costs $2000 How many of each item must they sell to buy the goals
3) A surf shop has a weekly overhead of $2300 How many skimboards and longboards must they sell to make a profit if skimboards are sold for $115 and longboards are sold for $685
4) Graph 7y + 42 lt 21x + 14 and shy3y lt 3x shy 12 on the same grid
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer121714
Write an equation of a line that is parallel to the line 2y = 3x shy 5 and goes through the point (4 shy3)
HW in the basket
Perpendicular Lines Lines that intersect at right angles
Slopes of perpendicular lines are negative reciprocals of each other
What is the slope of the line perpendicular to y = 2x shy 5
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) Write an equation in slopeshyintercept form for the line that passes through (shy4 6) and is perpendicular to the graph 3y = shy2x + 12
b) Write an equation in slopeshyintercept form for the line that passes through (47) and is perpendicular to the graph of 3y = 2x shy 1
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer111414
Write an equation parallel and a line perpendicular to 2x + 3y = 7
HW in the basket
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer
Write an equation of the line that goes through (3 4) and parallel to 2x + 3y = 10
Get a piece of graph paper
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer
Get a ruler and get ready for Test
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bonus
1) Write and graph the equation of a line parallel to the line x = 5
2) For her birthday Kwan received a $50 gift card to download songs The function m = shy050d + 50 represents the amount of money m that remains on the card after a number of songs d are downloaded Graph the equation and find how many songs she will be able to download
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
111814
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
1515Bell Ringer
Get a pencil ruler and a piece of graph paper
1) y = 2x shy 6
2) 3y = 9 shy 6x
3) 4x shy 5y = 15 shy x
4) 2x = 3y shy 7
1) 2 + 5k = 3k shy 6
2) 3w + 2 = 7w shy 5
3) 13c shy 6 = 33c + 28
4) Graph 4x + 2y = shy6
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
4 Question QuizWrite an equation of the line and graph
1) Through the point (3 4) and with a slope of shy2
2) Through the points (2 5) and (3 7)
3) Through the point (6 7) and parallel to 2y = 3x shy 8
4) Through the point (shy2 1) and perpendicular to y = shy2 x + 5
3
Bell Ringer
Describe the pattern amp what would the next three terms be
18 25 32
11215
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Topic 5
Sequences
A sequence is a _______ of numbers called the ________ of the sequence in a specific order
Ex 3 5 7 9 11
Since the difference between successive terms is ________ this is an arithmetic sequence
The difference between the terms is called the ________________ ________________ d
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) Find the common difference for the arithmetic sequence 33 29 25 21
b) Is the sequence 05 0625 075 08125 and arithmetic sequence
c) What is the next three terms in the sequence shy26 shy22 shy18 shy14
p[
Bell Ringer
Find the next three terms in the following sequences
813 18
32 27 22
11314
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
We can use a general formula to find any term in an arithmetic sequence given the first term and the common difference
Ex 8 11 14 17 Find the 50th term
a) Write an equation for the nth term of the arithmetic sequence shy12 shy8 shy4 0 Then find the 9th term
b) Write an equation for the nth term of the arithmetic sequence
3 shy10 shy23 shy36 Then find the 15th term
Which term of sequence is shy114
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer12015
Write the formula then find A32 and finally find n such that An = shy712
3 shy10 shy23
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Graphing the terms of an arithmetic sequence results in a ____________
The formula to find the nth term of an arithmetic sequence is the _____________ of the ________
a) Write and graph a function to represent the length of Martins long jumps
Jump 1 2 3 4
Distance (ft) 8 95 11 125
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer112014
Write the equation of the line that corresponds to the given arithmetic sequence
7 14 21
Bell Ringer
Place any homeworks in basket
Get ready for quiz
112414
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
BonusIn 1904 a dictionary cost $030 Since the cost of a dictionary has risen an average of $006 per year a) Write a linear equation to find the cost C of a dictionary y years after 1904b) If this trend continues what will the cost of a dictionary be in 2020
Bell Ringer12114
Diamond problem Find the product of the two numbers and place it in the top box find the sum and place it in the bottom box
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Topic 6
Inequalities
An open sentence that contains ____ _____ _____ or _____ is an __________
If the same number is added or subtracted to each side of a true inequality the resulting inequality is also true
If a gt b then a plusmn c gt b plusmn c
If a lt b then a plusmn c lt b plusmn c
Ex x shy 12 gt 8 m + 19 lt 56
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
SetshyBuilder Notationa more concise way of writing a solution set
Ex 1 x | x gt 20 Ex 2
a) p + 8 lt 18 b) 3a + 6 lt 4a
Verbal phrases
lt lt
gt gt
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Graphing one dimensional inequalities on a number line
Steps1) solve the given inequality for the variable
2) draw a number line with no numbers
3) place 0 in the center of the number line
4) place the numerical value you solved for on the number line
5) place the opposite of the numerical value on the opposite side of zero
6) draw a circle on the number you solved for
7) shade in the direction indicated by the inequality sign
General rules
Use an open circle with ______ or _______
Use a closed circle (shaded in) with ______ or _______
If the variable is on the left of the inequality then
shade to the left with ______ or _______
shade to the right with _______ or _______
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer12214 HW in the basket
Solve for the given variable and graph on a number linet + 15 lt shy24
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Graph the solution to
a) x shy 3 gt 7
c) 5 lt 7 + y
Solving multiplication and division problems involving inequalities
If the same positive number is multiplied or divided on both sides of a true inequality then the resulting inequality is also true
If the same negative number is multiplied or divided on both sides of a true inequality then the sign of the resulting inequality must be switched in order for it to be true
Ex) 4 gt 2 Ex) 7 lt 9
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) 30 gt n b) shy28 lt shy6g 2
c) shy3r gt 6 d) 9t lt 1084
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer12314 HW on your desk
Solve for the given variable and graph on a number lineshy5q gt shy45
HW Answers
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
MultishyStep Inequalities
Solved by undoing the operations the same way you would solve a multishystep equation
a) shy11y shy 13 gt 42 b) 23 gt 10 shy 2w
c) 43 gt shy4y + 11 d) 6(5z shy 3) lt 36z
e) Five minus 6 times a number is more than four times the number plus 45
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer12414 HW in the basket
Solve for the given variable and graph on a number line15 + 2h lt 4h + 27
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Compound Inequalities
When considered together two inequalities such as h gt 52 and h lt 72 form a compound inequality
A compound inequality containing and is only true if ____ inequalities are true
h gt 52
h lt 72
52 lt h lt 72
A compound inequality containing the word ____ is true if _________________of the inequalities is true
x gt 2
x lt shy1
x gt 2 x lt shy1
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
shy2 lt x shy 3 lt 4
HW pg 15632shy37 shy solve and graph41shy47 shy write an inequality
Take out a piece of paper and put your name on it(This is not a quiz)
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer12514 HW on your deskSolve for the given
variable and graph on a number line3 lt 4x shy 9 lt 7
HW on your desk
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Ex) The human ear can only detect sounds between the frequencies 20 Hertz and 20000 Hertz Write and graph a compound inequality that describes the frequency of sounds humans cannot hear
Ex) A company is manufacturing an action figure that must be at least 112 centimeters and at most 114 centimeters tall Write and graph a compound inequality that describes how tall the action figure can be
Homework
Study for Quiz
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Get a Ruler and a Pencil
7
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Get a colored pencil and ruler
Take out a sheet of graph paper and split it into four graphs
Graphing Inequalities in two variables
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality
An equation defines a _____________ which divides the coordinate plane into two ___________
The boundary may or may not be included in the solution When it is included the solution is a closed halfshyplanewhen it is not included the solution is an open halfshyplane
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Steps
1) Graph the boundary Use a solid line when the inequality contains ____ or _____Use a dashed line when the inequality contains _____ or ______
2) Use a test point to determine which halfshyplane should be shaded
3) Shade the halfshyplane that contains the solution
a) Graph 3x shy y lt 2 b) graph x + 5y gt 10
c) y gt x + 3 d) x shy y lt 3 2
e) Ranjan writes and edits short articles for a local newspaper It takes him about an hour to write an article and about a halfshyhour to edit an article If Ranjan works up to 8 hours a day how many articles can he write and edit in one day
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer1515
3
shy18
Solve for y
4x + 6y = 2x shy 30
Get a piece of graph paper
Topic 7
Linear Systems and Inequality Systems
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Two or more equations consisting of the same variables form a system of equations
A system can have one solution an infinite number of solutions or no solution
If a system has at least one solution it is said to be consistent
If there is exactly one solution it is said to be independent and consistent
If there is an infinite number of solutions it is said to be dependent and consistent
If a system has no solution it is said to be inconsistent The graphs are parallel
Number of Exactly one Infinite no solutionSolutions
Terminology Consistent and Consistent and InconsistentIndependent Dependent
Graph
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Solve a System of Linear Equations Graphically
Steps1) Graph both equations on the same set of axis
2) Circle and label the point of intersection if consistent and independent
3) Label whether it is consistent and independentdependent or inconsistent
Ex) y = shy2x + 2 and y = x shy 4
Bell Ringer1615
If x = (2r + 1) what is
3x + 6 in simplest terms
Take out the graphs from yesterday
HW on your desk 36
13
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) y = shy2x shy 5 and y = shy2x + 3
b) y = 2x + 4 and y = shy2x shy 2
c) y = shy2x shy 3 and 6x + 3y = shy9
HW Pg 415 numbers 3 6 9 12
HW Page 415 numbers 23shy26
3 (shy1 3)
6 (3 2)
9 (2 6)
12 (1 3)
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer1715
If x = (2y + 1) what is thevalue of y
3x + 2y = 27
HW in the basket
shy6
shy72
Solving a system of equations by substitution
Steps1) Solve at least one equation for one variable
2) Substitute the resulting expression from Step 1 into the other equation to replace the variable Then solve the equation
3) Substitute the value from Step 2 into either equation and solve for the other variable Write the solution as an ordered pair
Ex) y = 2x + 1 and 3x + y = shy9
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) y = 4x shy 6 and 5x + 3y = shy1
b) y = 2x shy 4 and shy6x + 3y = shy12
c) 2x shy y = 8 and y = 2x shy 3
Bell Ringer1815
HW on your desk
2
shy24
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Solving a system of equations using Elimination
Steps
1) Write the system so like terms with the same or opposite coefficients are aligned
2) Add or subtract the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 4x + 6y = 32 and 3x shy 6y = 3
a) shy4x + 3y = shy3 and 4x shy 5y = 5
b) 2t + 5r = 6 and 9r + 2t = 22
c) The sum of two numbers is shy10 Negative three times the first number minus the second number equals 2 Find the numbers
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer11215
HW from Thursday on your desk
CW from Friday in the basket
2
shy35
Elimination using Multiplication
Steps
1) Multiply at least one equation by a constant to get two equations that contain opposite terms
2) Add the equations eliminating one variable Then solve the equation
3) Substitute the value from Step 2 into one of the equations and solve for the other variable Write the solution as an ordered pair
Ex) 5x + 6y = shy8 and 2x + 3y = shy5
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) 6x shy 2y = 10 and 3x shy 7y = shy19
b) 9r + q = 13 and 3r + 2q = shy4
c) 4x + 2y = 8 and 3x + 3y = 9
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer11315
HW on your desk
11
24
Best Time to Use a Method
Method The Best Time to Use
Graphing To estimate solutions since graphing usually does not give an exact solution
Substitution If one of the variables in either equation has a coefficient of 1 or shy1
Elimination Using Addition If one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction If one of the variables has the same coefficient in the two equations
Elimination Using Multiplication If none of the coefficients are 1 or shy1 and neither of the variables can be eliminated by simply adding or subtracting the equations
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Determine the best method to solve the system of equations Then solve the system Use a calculator to check4x shy 4y = 8 and shy8x + y = 19
HW Page 425 numbers 15shy19
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer11415
HW on your desk shy36
0
Get a piece of graph paper
Solving a System of Inequalities
A set of two or more inequalities with the same variables is called a system of inequalities
Steps
1) Graph and label both inequalities
2) Place a large S where the two halfshyplanes intersect
3) Check a point in the solution
Ex) y gt shy2x + 1 and y lt x + 3
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
a) y lt 3 and x + y gt 1
b) 2x + y gt 2 and 2x + y lt 4
c) y gt shy4 and 3x + y lt 2
d) x + y gt 2 and shy4x + 2y lt 8
HW page 434 numbers 4 7 10 13
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Bell Ringer11515
HW on your desk54
21
Solving a system using matrices
4x + 2y = 83x + 3y = 9
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Solving using the Nspire
4x + 2y = 83x + 3y = 9
Bell Ringer11615
HW on your deskshy21
4
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Collected at end of class
Solve using substitution 2x shy 3y = shy2 and 4x + y = 24
Solve using elimination 3x + 4y = 6 and x + 2y = 4
Solve using elimination 4x shy 3y = 25 and shy3x + 8y = 10
Bell Ringer12015
Get a piece of graph paper56
15
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Unit 1 Espressions properties linear inequalities
Collected at the end of class
1) Solve graphically 4x shy 7y + 7 = 0 and 3x + 5y + 36 = 0
2) Solve graphically 2x + y lt 6 and x + y shy 2 gt 0
3) At a snack bar 3 pretzels and 1 can of soda cost $275 Two pretzels and 1 can of soda cost $200 Find the cost of a pretzel and the cost of a can of soda
4) Tickets for a high school dance cost $10 each if purchased in advance of the dance but $15 each if bought at the door If 100 tickets were sold and $1200 was collected how many tickets were sold in advance and how many were sold at the door
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
Attachments
Catholic Central High Schoolpdf
business_card_adsshy1shySCCCPDF
1o
c 2015 Yearbook Advertising Salest
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2015
Student Name
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Business Information
Business Name MQI Yiil 1 1 ` V ` L`
Contact Person f a U rv` l
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Business E- mail
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Patron Ad-$ 25Business Card ( 1 16 of a page) Ad-$ SO
of a Page Ad-$ 150
1 8 of a Page Ad-$ 100
Full Page Ad-$ 300
of a Page Ad-$ 200
Please attach the ad to this form and return to the Room 38 Ads can also be e- mailed to
yearbookcchstroy org please include the above information in the e- mail
Receipt Date
Business Name
Size of Business Ad Payment Check
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www1800LAW1010comAttorney Advertising Prior Results Do Not Guarantee a Similar Outcome
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- Attachments Page 1
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