solve for x. 30° 2x + 10 2x + 10 = 60 – 10 – 10 2x = 50 2 2 x = 25
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Bell Ringer
Solve for x.
30°
2x + 10
2x + 10 = 60 – 10 – 10 2x = 50
2 2 x = 25
Final Exam Review
Exam
The exam will be 40 multiple choice questions with 2 extra credit questions.
You will have 1 hour to complete the exam.
No extra time will be given. You may bring ONE sheet of notes to
use on the exam.
Topics
The final exam will cover:1. Inequalities2. Probability3. Area & Perimeter of Polygons & Circles4. Angles & Lines5. Exponents6. Radicals7. Polynomials
Inequalities
1. Graphing Linear Inequalities
< or > = Open Circle< or > = Closed Circle< or < = Shade to the left.> or > = Shade to the right.
.
.
.
Greater Than Less Than
1. Graphing Linear Inequalities
x < 3Open or Closed?
Right or Left?
x > -4Open or Closed?
Right or Left?
.
-5 + x < -1 + 5 +5
x < 4
Now graph it!
2. One-Step Linear Inequality
3. Two-Step Linear Inequality
4x + 1 > -11 – 1 – 1 4x > -12
4 4 x > -3
4. Reversing the Sign!
Whenever we multiply or divide by a negative number, we must REVERSE the inequality sign.
-2x < 6 -2 -2 x > -3
We have to divide
by -2.So we have to reverse the sign.
4. Reverse the Sign
-3x + 1 < 10 – 1 – 1
-3x < 9 -3 -3
x > -3Reverse the Sign!
Key Words
“At least”means greater than or equal to (>)
“No more than”means less than or equal to (<)
“More than”means greater than (>)
“Less than”means less than (<)
5. Word Problems
Chris has $200 in his bank account. He makes $10 an hour at his job. He wants to save at least $400 to buy some chickens. What is the minimum number of hours Chris will have to work?
200 + 10h > 400 – 200 – 200
10h > 200 10 10
h > 20 hours
Has already!
Wants more than
this amount!
Mo’ Money, mo’ money, mo’ money!
5. Word Problems
Tom calls a cab which charges $2.50 plus
$0.50 a mile. If Tom has no more than $20.00 in his pocket, how far can he go?
$2.50 + $0.50m < $20 -2.50 -2.50 0.50m < 17.50 0.50 0.50
m < 35 miles
That’s a lie. I got big bank!
Probability
Probability
Event – This is the selected outcome. Ex. If event A is the probability of rolling a 5 or higher, the probability is 2/7, so P(A) = 2/7.
Complement – This is the probability of everything other than the event.
Ex. In the example above, the complement is rolling 4 or lower, so the complement of event A is 5/7, or
P(A) = 5/7.Probability of “A Bar”
Coin Toss
If you toss a coin twice, what are the possible outcomes?
HH, TT, HT, TH What is the probability of two heads?
HH, TT, HT, TH =
What is the probability of at least one head?
HH, TT, HT, TH =
It’s complement would be 3/4!
It’s complement would be 1/4!
1/4
3/4
Two Independent Events
To find the probability of two independent events occurring together, multiply their probabilities!
Ex. Find the probability of tossing a coin twice and having heads occur twice.
12
12
. 14
=Probability of
Toss #1 coming up
heads.Probability of Toss #2
coming up heads.
Probability of two heads!
Independent Events
Ex. A coin is tossed and a card is drawn from a standard deck.a. What is the probability of tossing heads and drawing an ace?
b. What is the probability of tossing tails and drawing a face card?
12
113
. 126
=
12
313
. 326
=
Working with Factorials
4! + 3! =
3! – 2! =
4! 2! =
=
(4•3•2•1) + (3•2•1) = 30
(3•2•1) – (2•1) = 4
(4•3•2•1) (2•1) = 48
6!4!
(6•5•4•3•2•1) (4•3•2•1)
= 30
Combinations (Formula)
(Order doesn’t matter! AB is the same as BA)
nCr =
Where:n = number of things you can choose
fromr = number you are choosing
n!r! (n – r)!
Combinations (Formula)
There are 6 pairs of shoes in the store. Your mother says you can buy any 2 pairs. How many combination of shoes can you choose?
So n = 6 and r = 2
6C2 = = 6!2! (6 – 2)!
6•5•4•3•2•12•1(4•3•2•1)
=30 2 = 15 combinations!
Permutations (Formula)
(Order does matter! AB is different from BA)
nPr =
Where:n = number of things you can choose
fromr = number you are choosing
n! (n – r)!
Permutations (Formula)
In a 7 horse race, how many different ways can 1st, 2nd, and 3rd place be awarded?
So n = 7 and r = 3
7P3 = = 7!(7 – 3)!
7•6•5•4•3•2•1 (4•3•2•1)
= 210 permutations!
Fundamental Counting Principle
You have a choice of 3 meats, 4 cheeses, and 2 breads. How many different types of sandwiches could you make?
Multiply the choices!
3 • 4 • 2 = 24 different sandwiches
Area & Perimeter: Polygons and Circles
Key Vocabulary Perimeter – The distance around an
polygon.
Area – The amount of space inside a two dimensional shape.
Perimeter Estimate or calculate the length of
a line segment based on other lengths given on a geometric figure.
x
17 in
8 in
8 + x = 17 - 8 - 8 x = 9 in Easy!
Perimeter Compute the perimeter of polygons
when all side lengths are given
8 in
7 in
7 in
7 in
7 in8 in
Add all the sides:8 + 7 + 7 + 8 + 7 + 7 =
44 inEven I can do this!
Area Compute the area of rectangles
when whole number dimensions are given.
25 in
6 in
Area of Rectangle = Length • Width
A = L • WA = 25 • 6 A = 150 in2
Area Compute the area and perimeter of
triangles and rectangles in simple problems.
Area of Triangles = 1/2 • Base • Height
Base Base
Heig
ht
Heig
ht
Area Kyla mows lawns for $1.20 per
square feet. How much did she charge to cut the lawn below?
23 ft
11
ft
A = L • WA = 23 • 11A = 253 ft 2
Price = 253•$1.20Price = $303.60
Distance of Sides Find the missing value, x.
x
28 inx + 2x + x =
284x = 28
4 4x = 7 inches
I get it. Add up the bottom sides to equal the top!
x
2x
Perimeter Find the perimeter.
17 in
Find x:x + 5 =
17 – 5 – 5
x = 12 If x = 12, then x – 1
is 11!
x – 1
x + 5
Use x to find perimeter:17 + 17 + 11 + 11
Perimeter = 56 inches
I can find the area by cutting it!
Area Find the area.
5 ft
15 ftArea of ‘A’A = 10 • 7A = 70 ft2
Area of ‘B’A = 5 • 8A = 40 ft2
A = 70 + 40A = 110 ft2
8 ft
7 ft
5 ft1
0 f
t
AB
Area of Trapezoids Definition – Quadrilaterals with at least one
pair of parallel sides.
Area of a Trapezoid =
(Find the average of the bases and multiply by the height!)
b1
b2
h
b1 & b2 are the top and
bottom bases.
h is the height.
(b1+ b2) 2
• h
= • 9
Area of Trapezoids Find the area of the trapezoid below.
12
9
(b1+ b2) 2
A = • h
16
(12 + 16) 2
= 126 ft2
Parts of a Circle Circumference – The distance around a
circle. (Perimeter of a circle.) Radius – The distance from the center of a
circle to any point on its circumference. Diameter – The distance from one side of a
circle, passing through the center, to the other side of the circle.
.
Radius
Circumference Diameter
Circumference of a Circle The diameter of a circle is equal to
twice the radius, or
d = 2r Circumference of a circle is equal to
the diameter multiplied by pi, or
C = 2πr or
C = πd
Circumference of a Circle
C = 2πrC = 2 π 5C = 10π
or 31.42
5
Circumference of a Circle
C = dπC = 25.5π
or 80.1125.5
Diameter is 25.5!
Area of a Circle The area of a circle is equal to:
A = πr2.
6A = π62 A = 36πA = 113.10
Word Problems Pizza World offers two types of pizzas:
rectangles and circles. If each pizza cost $12.50, which is the better buy?
12 in
12
in
12 in diameter
A = 12 • 12= 144 in2
A = π • 62 = 113.10 in2
This is the
better buy!
Angles & Lines
Parallel Lines
You can identify parallel lines by their equations!
y = 3x + 7y = 3x – 9
These two lines are parallel. Their slopes are the same!
(Notice that they have different y-intercepts!)
Perpendicular Lines Lines that intersect at right angles (900)
are perpendicular. Perpendicular lines have slopes that are
negative reciprocals. The product of their slopes = -1.
These two lines are perpendicular. They
intersect at a right angle.
Perpendicular Lines
Negative reciprocals
1. What is the reciprocal of ?
2. What is the reciprocal of 3?
23
32
So the negative reciprocal is – ! 32
13
So the negative reciprocal is – ! 13
Perpendicular Lines
These equations are perpendicular:
y = 2x + 8
y = - x – 5
y = - x – 7
y = x + 5
12
45
32
54
YX
Complementary Angles
Complementary Angles – Two angles that add up to 90°
Angles X and Angle Y are
complementary and add up to 90
°.
Try this…
Find the missing angle.
36°
x°
90 – 36 = 54°
Supplementary Angles
Supplementary angles - Two angles that add up to 180°
Angles X and Angle Y are
supplementary and add up to
180 °.YX
Try this…
Solve for x.
x 138°
180 – 138 = 42°
Vertical Angles
Vertical Angles - A pair of opposite angles formed by the intersection of two lines. Vertical angles are always equal.
A
B
Angle A and Angle B are vertical angles. They are equal!
Corresponding Angles
nLine n is a transversal.
A
B
Corresponding angles – Two congruent angles that lie on the same side of the transversal.
A and B are corresponding angles.
Alternate Interior Angles
n
Line n is a transversal.
A
C
B
D
A = CÐB = D
They are alternate interior angles.
(Interior = Inside!)
Alternate Exterior Angles
n
Line n is a transversal.
A
C
B
D
A = CÐB = D
They are alternate exterior angles.
(Exterior = Outside!)
Try this…
Find the missing angles.
42° A
A = 138°B = 138°C = 42°
B C
FED
G
D = 42°E = 138°F = 138°G = 42°
Try this…
b°
d ° 65 °
70 ° 70 °
Hint: The 3 angles in a triangle sum to 180°.
Find the missing angles.
70 + 70 + b = 180140 + b = 180b = 40°
40 °
40 + 65 + d = 180105 + d = 180d = 75°
The Pythagorean Theorem
In a RIGHT Triangle, if sides “a” and “b” are the legs and side “c” is the hypotenuse, then
a2 + b2 = c2
a
b
c
Finding the Hypotenuse
Find the length of the hypotenuse.
12
16
C
a2 + b2 = c2
122 + 162 = c2
144 + 256 = c2
400 = c2
√400 = c20 = c
Finding the Leg
Find the length of the missing leg.
29
20
a
a2 + b2 = c2
a2 + 212 = 292
a2 + 400 = 841a2 = 441a = √441
a = 21
Exponents
Laws of Exponents
Zero Exponent Property – Any number raised to the zero power is 1.
x0 = 1 30 = 1 120 = 1Negative Exponent Property – Any number raised to a negative exponent is the reciprocal of the number.
x-4 = 5-1 = 5x-3y5 = 1x4
5y5
x3
15
Laws of Exponents
Product of Powers – When multiplying numbers with the same bases, ADD the exponents.
Quotient of Powers – When dividing numbers with the same bases, SUBTRACT the exponents.
x2•x8 = x10 3x4•x-2 = 3x2
x6
x2= x4 10x4y3
2x7y5y2
x3=
Laws of Exponents
(95)3 = 915
Power of a Power - When you have an exponent raised to an exponent, multiply the exponents!
(x4y5)2 = x8y10
Power of a Product - Distribute the exponent on the outside of the parentheses to all of the terms inside of the parentheses.
(xy)3 = x3y3 (2bc)2 = 4b2c2
Laws of Exponents
Power of a Quotient – Distribute the exponent on the outside of the parentheses to the numerator and the denominator of the fraction.
2x
=( )3 23
x3( )=8x3
3x2y
=( )4 34
x8y4( )= 81x8y4
Radicals
Simplifying Square Roots
√8 = √4 ∙ √2 = 2√2
8 is not a perfect
square, so we will
simplify it!
8 is made up of 4 ∙ 2. Look! 4 is a
perfect square!
√4 = 2We can’t
simplify √2, so we leave him alone.
Simplifying Square Roots
√75 =
√25 ∙ √3 =
5√3
Find factors of 75!
25 and 3 are factors of 75!
Find the square root of 25!
√72 =√36 ∙ √2 = 6√2
Try these…
√27
√32
√20
√75
= √9 ∙ √3
= √16 ∙ √2
= √4 ∙ √5
= √25 ∙ √3
= 3√3
= 4√2
= 2√5
= 5√3
Combining Square Roots
To combine square roots, combine the coefficients of like square roots.
4√3 + 5 √3= 9√3
7√5 – 4√2 =
Combine the coefficients, keep the radical!
You cannot combine unlike radicals!
7√5 – 4√2
Simplify and Combine√20 + √5 =√4 ∙ √5 + √5
=
√12 + √27 =
2√3
+√3 ∙ √43√3 =
√9 ∙ √3 =
5√3
2√5 + √5 =
3√5
+
Multiplying RadicalsWhen multiplying radicals, you can multiply the two numbers and put the answer under one radical. Simplify!
√3 ∙ √2 =
√6
√3 ∙ √3 =
√9 = 3
√3 ∙ √6 =
√18 = √9 ∙ √2
= 3√2
Multiplying Radicals
2√5 ∙ 3√5 1. Multiply the coefficients.
2 ∙ 3 = 62. Multiply the radicals.
√5 ∙ √5 = √253. Solve.
6√25 = 6 ∙ 5 = 30
Try This…
3√7 ∙ 2√5 =6√35
2√3 ∙ 5√3 =10√9 =10 ∙ 3
4√2 ∙ 3√8 =12√16= 12 ∙ 4
= 48
2√5 ∙ 3√2 =6√10
= 30
Polynomials
Add: (x2 + 3x + 1) + (4x2 +5)
Step 1: Identify like terms:
Step 2: Add the coefficients of like terms, do not change the powers of the variables:
Adding PolynomialsAdding Polynomials
(x2 + 3x + 1) + (4x2 +5)
Notice: ‘3x’ doesn’t have a like term.
(x2 + 4x2) + 3x + (1 + 5)
5x2 + 3x + 6
Subtract: (3x2 + 2x + 7) – (x2 + x + 4)
Subtracting PolynomialsSubtracting Polynomials
Step 1: Change subtraction to addition.
Step 2: Add like terms.
(3x2 + 2x + 7) + (- x2 + - x + - 4)
(3x2 + 2x + 7)+ (- x2 + - x + - 4)
2x2 + x + 3
Change signs of all terms after subtraction sign.
Multiplying Polynomials
Distributive Property – Review!!!
3(x + 5) = 3x + 15
We will need that same concept to multiply polynomials.
3(2x2 + 4x + 3)= 6x2
We will distribute the outside term to everything on the inside.
+ 12x
+ 9
Multiplying Polynomials
4(3x2 – 2x + 1) = 12x2 – 8x + 4
2x(2x2 + x + 5) = 4x3 + 2x2 + 10x
4 ∙ 3x2 4 ∙ –2x
4 ∙ 1
2x ∙ 2x2
2x ∙ x
2x ∙ 5
12x2 –8x +
4
4x3 +2x2
+10x
Multiplying Polynomials: Expansion Boxes VS. FOIL
(2x + 1)(4x – 3)
Combine:8x2 – 6x + 4x – 3
8x2 – 2x – 3
Mult. 4x – 3
2x
+ 1
8x2 – 6x
+ 4x
– 3
First: +2x ∙ +4x = +8x2
Outer: +2x ∙ -3 = -6xInner: +1 ∙ +4x = +4xLast: +1 ∙ -3 = -3Combine:
8x2 – 6x + 4x – 3 8x2 – 2x – 3
Same Answer!Same Answer!
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