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ALGEBRA 1Teacher:

Unit 2Chapter 3Chapter 4Chapter 5

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UPDATED FALL 2015Algebra 1Section 3.3 Notes: Rate of Change and Slope

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Warm-UpEvaluate:

1) 2−43−1

2) 3−(−5)10−6

3) 6−7

3−(−3)4)

6−(−6)10−1

Rate of Change: a that describes, on average, how much a quantity changes with respect to a change in another quantity.

Example 1: Use the table to find the rate of change. Then explain its meaning.

In example 1 the rates of change have been constant. Many real-world situations involve rates of change that are _______ constant.

Example 2: The graph below shows the number of U.S. passports issued in 2002, 2004, and 2006.

a) find the rates of change for 2002- 2004 and 2004 – 2006.

b) Explain the meaning of the rate of change in each case.

c) How are the different rates of change shown on the graph?

A rate of change is constant for a function when the rate of change is the ____________ between any pair of points on the graph of the function. functions have a constant rate of change.

Example 3: Determine whether each function is linear.

a) b)

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Slope: the _____________ of the change in the y – coordinates (______________) to the change in the x – coordinates (_________) as you move from one point to another.

Slope describes how ___________ a line is. The _____________ the absolute value of the slope, the ________________ the line.

Because a linear function has a constant rate of change, __________________ on a nonvertical line can be used to determine its slope.

The slope of a line can be ____________________, ___________________, _______________, or ______________________.

If the line is not horizontal or vertical, then the slope is either positive or negative.

Example 4: Find the slope of a line that passes through each pair of points.

a) (-3, 2) and (5, 5) b) (-3, -4) and (-2, -8) c) (-3, 4) and (4, 4)

Example 5: Find the slope of the line that passes through (-2, -4) and (-2, 3).

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3.3 Day 1 Textbook Homework

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Section 3.3 DAY 2 Notes: Slope and Rate of ChangeWarm-up

1) Find the slope between the points (-5, 8) and (-10, - 3)

2) Write two points that a line passing through would have undefined slope.

3) Write two points that a line passing through would have zero slope.

Example 6: Find the value of r so the line passes through each pair of points and has the given slope.

a) (6, 3), (r, 2); m=12 b) (–2, 6), (r, – 4); m=−5 c) (r, –6), (5, –8); m=−8

Example 7: Use the graph to approximate the average rate of change between x = - 1 and x = 3.

Example 8:Which table shows golf scores that decrease by the same amount each game?

A) Game Score

1 922 883 82

B) Game Score

1 722 693 65

C)Game Score

1 792 763 73

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Algebra 1Section 3.3 Worksheet

Find the slope of the line that passes through each pair of points.

1. 2. 3.

4. (6, 3), (7, –4) 5. (–9, –3), (–7, –5) 6. (6, –2), (5, –4)

7. (7, –4), (4, 8) 8. (–7, 8), (–7, 5) 9. (5, 9), (3, 9)

10. (15, 2), (–6, 5) 11. (3, 9), (–2, 8) 12. (–2, –5), (7, 8)

13. (12, 10), (12, 5) 14. (0.2, –0.9), (0.5, –0.9) 15. ( 73

, 43 ) , (−1

3, 23 )

Find the value of r so the line that passes through each pair of points has the given slope.

16. (–2, r), (6, 7), m = 12

17. (–4, 3), (r, 5), m = 14

18. (–3, –4), (–5, r), m = – 92

19. (–5, r), (1, 3), m = 76

20. (1, 4), (r, 5), m undefined 21. (–7, 2), (–8, r), m = –5

22. (r, 7), (11, 8), m = – 15

23. (r, 2), (5, r), m = 0

24. ROOFING The pitch of a roof is the number of feet the roof rises for each 12 feet horizontally. If a roof has a pitch of 8, what is its slope expressed as a positive number?

25. SALES A daily newspaper had 12,125 subscribers when it began publication. Five years later it had 10,100 subscribers. What is the average yearly rate of change in the number of subscribers for the five-year period?

26. HIGHWAYS Roadway signs such as the one below are used to warn drivers of an upcoming steep down grade that could lead to a dangerous situation. What is the grade, or slope, of the hill described on the sign?

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27. AMUSEMENT PARKS The SheiKra roller coaster at Busch Gardens in Tampa, Florida, features a 138-foot vertical drop. What is the slope of the coaster track at this part of the ride? Explain.

28. CENSUS The table shows the population density for the state of Texas in various years. Find the average annual rate of change in the population density from 2000 to 2009.

Population Density

Year People Per Square Mile

1930 22.1

1960 36.4

1980 54.3

2000 79.6

2009 96.7 Source: Bureau of the Census, U.S. Dept. of Commerce

29. REAL ESTATE A realtor estimates the median price of an existing single-family home in Cedar Ridge is $221,900. Two years ago, the median price was $195,200. Find the average annual rate of change in median home price in these years.

30. COAL EXPORTS The graph shows the annual coal exports from U.S. mines in millions of short tons.

Source: Energy Information Association

a. What was the rate of change in coal exports between 2001 and 2002?

b. How does the rate of change in coal exports from 2005 to 2006 compare to that of 2001 to 2002?

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c. Explain the meaning of the part of the graph with a slope of zero.

Algebra 1Section 4.3 Notes: Writing Equations in Point-Slope Form Day 1

Warm-Up

Find the slope of the line that has a x – intercept of – 2 and a y – intercept of 5.

Point-slope form: an equation that can be written in the form __________________________where ___________ is the slope and ___________________ is a given point.

Example 1: Write an equation in point-slope form for the line that passes through (– 2, 0) with a slope of −32

.

If you are given the slope and the coordinates of one or two points, you can write the linear equations in the following ways.

Given Slope and One Point:

Step 1: ________________________ the value of ______and let the x and y coordinates by ___________________.

Step 2: ___________________________________________________.

Example 2: Write an equation using point – slope form for each line using the given information.

a.Slope is 1, Point is (4, 6) b. Slope is -4, Point is (-2, 8)

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c. Slope is 12

,Point is (3, -2) d. Slope is 23

, passes through (0, 2)

e. Slope is 0, Point is (2, -5) e. Slope is 1, passes through (-4, 0)

Given Two Points:

Step 1: _______________________________________________________.

Step 2: Choose __________________________ of the points to use.

Step 3: Follow the steps for writing an equation given the ____________________________________.

Example 3: Write an equation using point – slope form for a line passing through the 2 points given.

a. (- 3, 2) and (4, 7) b. (6, - 3) and (10, 1)

Example 4: The figure shows trapezoid ABCD, with bases AB and CD

Write an equation in point-slope form for the line containing the side BC.

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Algebra 1Section 4.3 Notes: Writing Equations in Point-Slope Form Day 2 (Standard Form)

Warm – UpWrite an equation in point – slope form for the line with the given information.

1) (5, - 5), m = 2 2. ( - 1, 5), m = −716

Standard form: ______________________ where A ≥ 0, A and B are not both zero, and A, B, and C are integers with a greatest common factor of 1.

Example 1: Write the following equation in standard form.

a) y=34

x−5 b) y−1=7(x+5)

Writing an Equation in Standard Form Given Two Points

Step 1: Find the __________________________________ between the two points.

Step 2: Plug in the value of ______ and then choose either point to plug in as (x1, y1) into point – slope form. [__________________]

Step 3: Rewrite the equation into standard form. [_________________________]

Example 2: Write an equation in standard form that passes through the given points.

a) ( - 4, 2) and (3, 8) b) x – intercept: (3, 0) and y – intercept (0, - 5)

c) (−7 ,−3) and (−3 , 5) d) (−1 ,3) and y-intercept of 8.

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Example 3: Which equation has a graph with a slope of 2 and a y-intercept of 9?A. y - 9 = 2xB. y - 9 = -2xC. y = 2(x - 9)D. y = 2(x + 9)

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Write an equation in point-slope form for the line that passes through each point with the given slope.

1. (2, 2), m = –3 2. (1, –6), m = –1 3. (–3, –4), m = 0

4. (1, 3), m = −34

5. (–8, 5), m = −25

6. (3, –3), m = 13

Write each equation in standard form.

7. y – 11 = 3(x – 2) 8. y – 10 = –(x – 2) 9. y + 7 = 2(x + 5)

10. y – 5 = 32

(x + 4) 11. y + 2 = −34

(x + 1) 12. y – 6 = 43

(x – 3)

13. y + 4 = 1.5(x + 2) 14. y – 3 = –2.4(x – 5) 15. y – 4 = 2.5(x + 3)

Write each equation in slope-intercept form.

16. y + 2 = 4(x + 2) 17. y + 1 = –7(x + 1) 18. y – 3 = –5(x + 12)

19. y – 5 = 32

(x + 4) 20. y – 14

= – 3(x + 14

) 21. y – 23

= –2(x – 14

)

22. CONSTRUCTION A construction company charges $15 per hour for debris removal, plus a one-time fee for the use of a trash dumpster. The total fee for 9 hours of service is $195.

a. Write the point-slope form of an equation to find the total fee y for any number of hours x.

b. Write the equation in slope-intercept form.

c. What is the fee for the use of a trash dumpster?

23. MOVING There is a daily fee for renting a moving truck, plus a charge of $0.50 per mile driven. It costs $64 to rent the truck on a day when it is driven 48 miles.

a. Write the point-slope form of an equation to find the total charge y for a one-day rental with x miles driven.

b. Write the equation in slope-intercept form.

c. What is the daily fee?

24. BICYCLING Harvey rides his bike at an average speed of 12 miles per hour. In other words, he rides 12 miles in 1 hour, 24 miles in 2 hours, and so on. Let h be the number of hours he rides and d be distance traveled. Write an equation for the relationship between distance and time in point-slope form.

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25. GEOMETRY The perimeter of a square varies directly with its side length. The point-slope formof the equation for this function is y – 4 = 4(x – 1). Write the equation in standard form.

26. NATURE The frequency of a male cricket’s chirp is related to the outdoor temperature. The relationship is expressed by the equation T = n + 40, where T is the temperature in degrees Fahrenheit and n is the number of chirps the cricket makes in 14 seconds. Use the information from the graph below to write an equation for the line in point-slope form.

27. CANOEING Geoff paddles his canoe at an average speed of 3.5 miles per hour. After 5 hours of canoeing, Geoff has traveled 18 miles. Write an equation in point-slope form to find the total distance y for any number of hours x.

28. AVIATION A jet plane takes off and consistently climbs 20 feet for every 40 feet it moves horizontally. The graph shows the trajectory of the jet.

a. Write an equation in point-slope form for the line representing the jet’s trajectory.

b. Write the equation from part a in slope -intercept form.

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c. Write the equation in standard form.Algebra 1Section 4.2 Notes: Writing Equations in Slope-Intercept Form

Warm-UpThe figure shows parallelogram ABCD. a) Write an equation in point – slope form of side BC.

b) Write an equation in standard form of side BC.

Write an equation in slope – intercept form given slope and point:

Step 1: Plug in _____________________________________________________________________________

Step 2: Solve the equation for ________.

Example 1:

a) Write an equation of a line that passes through (2, – 3) with a slope of 12

.

b) Write an equation of a line that passes through (–2, 5) with a slope of 3.

If you are given two points through which a line passes, you can use them to find _________________. Then follow the steps above.

Example 2: Write an equation of the line that passes through each pair of points.

a) (–3, –4) and (–2, –8) b) (6, –2) and (3, 4)

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Constraint: a condition that a solution must ____________________.

Equations can be viewed as constraints in a problem situation. The solutions of the equation meet the constraints of the problem.

Example 3: During one year, Malik’s cost for self-serve regular gasoline was $3.20 on the first of June and $3.42 on the first of July. Write a linear equation to predict Malik’s cost of gasoline the first of any month during the year, using 1 to represent January.

Linear extrapolation: the use of a linear equation to ________________ values that our outside the range of data.

Example 4: On average, Malik uses 25 gallons of gasoline per month. He budgeted $100 for gasoline in October. Use the prediction equation in Example 3 to determine if Malik will have to add to his budget. Explain.

Example 5: Which situation can be modeled by the equation y = 3x + 16?

A) You have a summer job where you earn $3 per day cleaning cars, but earn an additional $16 per car cleaned during your shift. Let y equal your wages earned in one day.

B) You have a summer job where you earn $16 per day cleaning cars but earn and additional $3 per car cleaned during your shift. Let y equal your wages earned in one day.

C) You make a $3 payment on a new phone, and then will be $16 per month until the phone is paid off. Let y equal the total amount you will pay for the phone.

D) You have $3 in your pocket, and want to buy a pizza that costs $16. Let y equal the total amount you need to buy the pizza.

Example 6: The equation of a line is given in standard form. Find the slope of each line.

a) 4x + 3y = 7 b) 3x + 2y = 10

Example 7: The equation C = 0.67t + 1.30 represents the cost C of a burger with t toppings. Which statement is true?

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A) Each topping costs $1.30.B) A burger with no topping costs $1.30.C) A burger with 3 toppings costs $1.97.D) A burger with no toppings costs $0.67.4.2 Textbook Homework

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Write an equation of the line that passes through the given point and has the given slope.

1. 2. 3.

4. (–5, 4); slope –3 5. (4, 3); slope 12

6. (1, –5); slope −32

7. (3, 7); slope 27

8. (−2 , 52 ) ; slope

−12

9. (5, 0); slope 0

Write an equation of the line that passes through each pair of points.

10. 11. 12.

13. (0, –4), (5, –4) 14. (–4, –2), (4, 0) 15. (–2, –3), (4, 5)

16. (0, 1), (5, 3) 17. (–3, 0), (1, –6) 18. (1, 0), (5, –1)

19. DANCE LESSONS The cost for 7 dance lessons is $82. The cost for 11 lessons is $122. Write a linear equation to find the total cost C for ℓ lessons. Then use the equation to find the cost of 4 lessons.

20. WEATHER It is 76°F at the 6000-foot level of a mountain, and 49°F at the 12,000-foot level of the mountain. Write a linear equation to find the temperature T at an elevation x on the mountain, where x is in thousands of feet.

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21. FUNDRAISING Yvonne and her friends held a bake sale to benefit a shelter for homeless people. The friends sold 22 cakes on the first day and 15 cakes on the second day of the bake sale. They collected $88 on the first day and $60 on the second day. Let x represent the number of cakes sold and y represent the amount of money made. Find the slope of the line that would pass through the points given.

22. JOBS Mr. Kimball receives a $3000 annual salary increase on the anniversary of his hiring if he receives a satisfactory performance review. His starting salary was $41,250. Write an equation to show k, Mr. Kimball’s salary after t years at this company if his performance reviews are always satisfactory.

23. CENSUS The population of Laredo, Texas, was about 215,500 in 2007. It was about 123,000 in 1990. If we assume that the population growth is constant and t represents the number of years after 1990, write a linear equation to find p, Laredo’s population for any year after 1990.

24.WATER Mr. Williams pays $40 a month for city water, no matter how many gallons of water he uses in a given month. Let x represent the number of gallons of water used per month. Let y represent the monthly cost of the city water in dollars. What is the equation of the line that represents this information? What is the slope of the line?

25. SHOE SIZES The table shows how women’s shoe sizes in the United Kingdom compare to women’s shoe sizes in the United States.

a. Write a linear equation to determine any U.S. size y if you are given the U.K. size x.

Source: DanceSport UK

b. What are the slope and y-intercept of the line?

c. Is the y-intercept a valid data point for the given information?

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Women’s Shoe Sizes

U.K. 3 3.5 4 4.5 5 5.5 6

U.S. 5.5 6 6.5 7 7.5 8 8.5

Algebra 1 Name:

Review 3.3, 4.2, 4.3 Period:

Write down the following formula and equation form.

Slope Formula: Point-Slope Form:

Standard Form: Slope-Intercept Form:

1. There were 526 million Twitter users in 2008. The amount of users increased to 974 million Twitter users in the beginning of 2014. What was the rate of change of Twitter users from 2008 to 2014?

Find the slope of the line that passes through Find the value of r so the line that passes through eacheach pair of points. pair of points has the given slope.2. (−2 , 4 ) ,(0 ,−1) 3. (6 , r ) , (7 ,−4 ) , m=−7

Write the equation in Point-Slope Form.4. Given (0 ,−1), Slope is 3 5. Given (6 ,−2 ) ,(5 ,−4) 6. Given (5 ,9 ) ,(3 ,9)

Write the equation in Standard Form.

7. y=−23

x+4 8. y−3=14(x+3) 9. Given (−2 , 3 ) ,(−1 , 0)

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Write the equation in Slope-Intercept Form.10. Given slope is −1 and y-intercept is 3 11. Given (−3 ,5) and a slope of −2

12. Given (4 ,−2 ) ,(2 ,−4) 13. Given

14. At Momma’s Little Bakery, 6 doughnuts cost $7. The cost for 24 doughnuts is $28. Write a linear equation to find the total cost C of d doughnuts. Then use the equation to find the cost of a dozen doughnuts.

15. Each week, Derrick gets an allowance of $5 when he completes all of his chores. He already saved $46 when his parents decided to start giving him an allowance. Write an equation to show T, Derrick’s total amount saved after w weeks of allowance. What is the slope of this line?

Equation: Slope:

16. Using the points (1 ,0 ) and (5 ,−1), write the equation of the line in the following forms:

Point-Slope Form:

Slope-Intercept Form:

Standard Form:

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X Y-3 1-1 -3

Algebra 1Section 4.1 Notes: Graphing Equations in Slope-Intercept Form

Slope-intercept form: an equation of the form ________________, where m is the ________ and b is the ______________.

The variables m and b are called ________________ of the equation. Changing either value changes the equation’s graph.

When graphing an equation.

1) Plot the ______________________.

2) Use the slope to ____________ additional points.

3) _____________ a line through the points.

Example 1: Write an equation in slope intercept form of the line with a slope of 14

and a y-intercept of – 1. Then graph the equation.

When an equation is not written in slope-intercept form, it may be easier to _________________ before graphing.

Example 2: Graph 5 x+4 y=8

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Constant Functions: a linear function of the form _____________.

Constant functions do not cross the ____________ except when y=0. Constant functions are ________________ lines therefore their slope is ________. Their domain is _________________, and their range is ___________.

Vertical lines have _______________. So, equations of vertical lines __________________________________________.

Example 3: a) Graph y=−7 b) Graph x = 2

Writing an equation given a graph.

1) Locate the ________________________

2) Find the slope by using ________________________ to find another point on the graph.

3) ___________________________________ in slope-intercept form

Example 4: Write an equation in slope-intercept form for the line shown in the graph.

Example 5: The ideal maximum heart rate for a 25 year old exercising to burn fat is 117 beats per minute. For every five years older than 25, that ideal rate drops three beats per minute.

a) Write a linear equation to find the ideal maximum heart rate for anyone over 25 who is exercising to burn fat.

b) Graph the equation.

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c) Find the ideal maximum heart rate for a 55 year old person exercising to burn fat. Example 6: The graph of a linear function is shown below. Which graph shows the same function with its y-intercept changed to 3?

A) B) C) D)

Example 7: What is the equation of the line graphed?

a) b)

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4.1 Textbook Homework

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Write an equation of a line in slope-intercept form with the given slope and y-intercept.

1. slope: 14

, y-intercept: 3 2. slope: 32

, y-intercept: –4

3. slope: 1.5, y-intercept: –1 4. slope: –2.5, y-intercept: 3.5

Write an equation in slope-intercept form for each graph shown.

5. 6. 7.

Graph each equation.

8. y = −12

x + 2 9. 3y = 2x – 6 10. 6x + 3y = 6

11. WRITING Carla has already written 10 pages of a novel. She plans to write 15 additional pages per month until she is finished.

a. Write an equation to find the total number of pages P written after any number of months m. Carla’s Novel

b. Graph the equation on the grid at the right.

c. Find the total number of pages written after 5 months.

12. SAVINGS Wade’s grandmother gave him $100 for his birthday. Wade wants to save his money to buy a new MP3 player that costs $275. Each month, he adds $25 to his MP3 savings. Write an equation in slope-intercept form for x, the number of months that it will take Wade to save $275.

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13. CAR CARE Suppose regular gasoline costs $2.76 per gallon. You can purchase a car wash at the gas station for $3. The graph of the equation for the cost of x gallons of gasoline and a car wash is shown below. Write the equation in slope-intercept form for the line.

14. ADULT EDUCATION Angie’s mother wants to take some adult education classes at the local high school. She has to pay a one-time enrollment fee of $25 to join the adult education community, and then $45 for each class she wants to take. The equation y = 45x + 25 expresses the cost of taking x classes. What are the slope and y-intercept of the equation?

15. BUSINESS A construction crew needs to rent a trench digger for up to a week. An equipment rental company charges $40 per day plus a $20 non-refundable insurance cost to rent a trench digger. Write and graph an equation to find the total cost to rent the trench digger for d days.

16. ENERGY From 2002 to 2005, U.S. consumption of renewable energy increased an average of 0.17 quadrillion BTUs per year. About 6.07 quadrillion BTUs of renewable power were produced in the year 2002.

a. Write an equation in slope-intercept form to find the amount of renewable power P (quadrillion BTUs) produced in year y between 2002 and 2005.

b. Approximately how much renewable power was produced in 2005?

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c. If the same trend continues from 2006 to 2010, how much renewable power will be produced in the year 2010Algebra 1Section 5.6 Notes: Graphing Inequalities in Two Variables

Warm-UpWrite an equation in slope-intercept form of each line.

a) slope is 3; y – intercept is – 1 b) slope is −45

; y – intercept is 0

Example 1: Determine which ordered pairs in the set are a part of the solution to the inequality.{(-2,-2), (1,-1), (1,1), (2,5), (6,0)}

Interceptsx-intercept:

Place where graph crosses _____________________. Always in the form of _____________________. Can be found by ________________________________________________________________.

y-intercept: Place where graph crosses _____________________. Always in the form of _____________________. Can be found by ________________________________________________________________.

Example 2: Find the x and y intercepts of

Graphing in Standard Form

Steps for Graphing an Equation in Standard Form ( )1. Find the _____________________________ by plugging in 0 for y and solving for x.

2. Find the _____________________________by plugging in 0 for x and solving for y.

3. Plot the two intercepts on the graph and connect the points to form a line.

Example 3: Graph the equation

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http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&docid=GyO0HDlb4fB2lM&tbnid=jnr2X-RTGYeE8M:&ved=0CAUQjRw&url=http://teachers.mpcsd.org/mdoroquez/pages/Alg.htm&ei=dEOWU7S5NIaTyATY8IKgAQ&bvm=bv.68445247,d.aWw&psig=AFQjCNFIzmAGEIkGX95aJoFQxlhyXiy2Ww&ust=1402442953457026

Example 4: Which is the correct graph of 2x + 2y = -4?

A) B) C) D)

The graph of a linear inequality is the ____________________________ that represent _____________________________________

___________________________________.

An equation defines a ________________________, which divides the coordinate plane into _____________________________.

The boundary may or not be included in the solution. When it is included, the solution is a _________________________ (solid line).

When not included, the solution is an _______________________ (dashed line).

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Example 5: Graph the inequality

a) 2 y−4 x>6 b) x−1> y

Example 6: Graph the inequality.

a) x+4 y≥ 2 b) 2 x+3 y ≥18

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You can use a coordinate plane to solve inequalities with one variable.

Example 7: Use a graph to solve.

a) 2 x+3≤ 7 b) −2 x+6>12

An inequality can be viewed as a constraint in a problem situation. Each solution of the inequality represents a combination that meets the constraint. In real-world problems, the domain and range are often restricted to nonnegative or whole numbers.

Example 8: Write, Solve and Graph an Inequality

a) Ranjan writes and edits short articles for a local newspaper. It takes him about an hour to write an article and about a half-hour to edit an article. If Ranjan works up to 8 hours a day, how many articles can he write and edit in one day?

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b) Neil wants to run a marathon at a pace of at least 6 miles per hour. Write and graph an inequality for the miles y he will run in x hours.

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Determine which ordered pairs are part of the solution set for each inequality.

1. 3x + y ≥ 6, {(4, 3), (–2, 4), (–5, –3), (3, –3)}

2. y ≥ x + 3, {(6, 3), (–3, 2), (3, –2), (4, 3)}

3. 3x – 2y < 5, {(4, –4), (3, 5), (5, 2), (–3, 4)}

Graph each inequality.

4. 2y – x < –4 5. 2x – 2y ≥ 8 6. 3y > 2x – 3

Use a graph to solve each inequality.

7. –5 ≤ x – 9 8. 6 > 23

x + 5 9. 12

> –2 x + 72

10. MOVING A moving van has an interior height of 7 feet (84 inches). You have boxes in 12 inch and 15 inch heights, and want to stack them as high as possible to fit. Write an inequality that represents this situation.

11. BUDGETING Satchi found a used bookstore that sells pre-owned DVDs and CDs. DVDs cost $9 each, and CDs cost $7 each. Satchi can spend no more than $35.

a. Write an inequality that represents this situation.

b. Does Satchi have enough money to buy 2 DVDs and 3 CDs?

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12. FAMILY Tyrone said that the ages of his siblings are all part of the solution set of y > 2x, where x is the age of a sibling and y is Tyrone’s age. Which of the following ages is possible for Tyrone and a sibling?Tyrone is 23; Maxine is 14.Tyrone is 18; Camille is 8.Tyrone is 12; Francis is 4.Tyrone is 11; Martin is 6.Tyrone is 19; Paul is 9.

13. FARMING The average value of U.S. farm cropland has steadily increased in recent years. In 2000, the average value was $1490 per acre. Since then, the value has increased at least an average of $77 per acre per year. Write an inequality to show land values above the average for farmland.

14. SHIPPING An international shipping company has established size limits for packages with all their services. The total of the length of the longest side and the girth (distance completely around the package at its widest point perpendicular to the length) must be less than or equal to 419 centimeters. Write and graph an inequality that represents this situation.

15. FUNDRAISING Troop 200 sold cider and donuts to raise money for charity. They sold small boxes of donut holes for $1.25 and cider for $2.50 a gallon. In order to cover their expenses, they needed to raise at least $100. Write and graph an inequality that represents this situation.

16. INCOME In 2006 the median yearly family income was about $48,200 per year. Suppose the average annual rate of change since then is $1240 per year.

a. Write and graph an inequality for the annual family incomes y that are less than the median for x years after 2006.

b. Determine whether each of the following points is part of the solution set.

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(2, 51,000) (8, 69,200)

(5, 50,000) (10, 61,000)Algebra1 1Section 4.4 Notes: Parallel and Perpendicular Lines

Warm – UpWhich is the correct graph of x – y < 5?

A) B) C) D)

Parallel lines: lines in the same plane that .

Finding the equation of a line given the equation of a parallel line and a point on the line

1. Find the of the given line.

2. Substitute the provided and the from the given line into ______.

Example 1:

a) Write an equation in slope-intercept form for the line that passes through (4, – 2) and is parallel to y=12

x−7.

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b) Write an equation in point-slope form for the line that passes through (4, – 1) and is parallel to y= 14

x+7.

Perpendicular lines: lines that intersect at _______angles.

The slopes of nonvertical perpendicular lines are .

You can use slope to determine whether two lines are perpendicular.

You can determine whether the graphs of two linear equations are parallel or perpendicular by comparing the __________ of the lines.

Example 2:

a) Determine whether the graphs of 3 x+ y=12, y=13

x+2, and 2 x−6 y=−5 are parallel or perpendicular. Explain.

b) Determine whether the graphs of 6 x−2 y=−2, y=3 x−4, and y=4 are parallel or perpendicular. Explain.

You can write the equation of a line perpendicular to a given line if you know a point on the line and the equation of the given line.

Example 3:

a) Write an equation in slope-intercept form for the line that passes through (4, – 1) and is perpendicular to the graph of 7 x−2 y=3.

b) Write an equation in slope-intercept form for the line that passes through (4, 7) and is perpendicular to the graph of y=23

x−1.

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49


Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the graph of the given equation.

1. (3, 2), y = x + 5 2. (–2, 5), y = –4x + 2 3. (4, –6), y = −34

x + 1

4. (5, 4), y = 25

x – 2 5. (12, 3), y = 43

x + 5 6. (3, 1), 2x + y = 5

7. (–3, 4), 3y = 2x – 3 8. (–1, –2), 3x – y = 5 9. (–8, 2), 5x – 4y = 1

10. (–1, –4), 9x + 3y = 8 11. (–5, 6), 4x + 3y = 1 12. (3, 1), 2x + 5y = 7

Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of the given equation.

13. (–2, –2), y = −13

x + 9 14. (–6, 5), x – y = 5 15. (–4, –3), 4x + y = 7

16. (0, 1), x + 5y = 15 17. (2, 4), x – 6y = 2 18. (–1, –7), 3x + 12y = –6

19. (–4, 1), 4x + 7y = 6 20. (10, 5), 5x + 4y = 8 21. (4, –5), 2x – 5y = –10

22. (1, 1), 3x + 2y = –7 23. (–6, –5), 4x + 3y = –6 24. (–3, 5), 5x – 6y = 9

25. GEOMETRY Quadrilateral ABCD has diagonals AC and BD. Determine whether AC is perpendicular to BD . Explain.

26. GEOMETRY Triangle ABC has vertices A(0, 4), B(1, 2), and C(4, 6). Determine whether triangle ABC is a right triangle. Explain.

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27. BUSINESS Brady’s Books is a retail store. The store’s average daily profits y are given by the equation y = 2x + 3 where x is the number of hours available for customer purchases. Brady’s adds an online shopping option. Write an equation in slope-intercept form to show a new profit line with the same profit rate containing the point (0, 12).

28. ARCHITECTURE The front view of a house is drawn on graph paper. The left side of the roof of the house is represented by the equation y = x. The rooflines intersect at a right angle and the peak of the roof is represented by the point (5, 5). Write the equation in slope-intercept form for the line that creates the right side of the roof.

29. ARCHAEOLOGY An archaeologist is comparing the location of a jeweled box she just found to the location of a brick wall. The

wall can be represented by the equation y = −53

x + 13. The box is located at the point (10, 9). Write an equation representing a

line that is perpendicular to the wall and that passes through the location of the box.

30. GEOMETRY A parallelogram is created by the intersections of the lines x = 2, x = 6, y = 12

x + 2, and another line. Find the

equation of the fourth line needed to complete the parallelogram. The line should pass through (2, 0). (Hint: Sketch a graph to help you see the lines.)

31. INTERIOR DESIGN Pamela is planning to install an island in her kitchen. She draws the shape she likes by connecting the vertices of the square tiles on her kitchen floor. She records the location of each corner in the table.

CornerDistance

from WestWall (tiles)

Distancefrom SouthWall (tiles)

A 5 4

B 3 8

C 7 10

D 11 7a. How many pairs of parallel sides are there in the shape ABCD she designed? Explain.

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b. How many pairs of perpendicular sides are there in the shape she designed? Explain.

c. What is the shape of her new island?

Algebra 1Section 4.7 Notes: Inverse Linear Functions

Warm-Up

1. Find the inverse of f ( x )=−3 x+4 2. Find the inverse of f ( x )=12(x−3)

Inverse relation: the set of ordered pairs obtained by ______________________________________of each ordered pair in a relation.

Notice the domain of a relation becomes the _____________________________________________, and the range of the relation

becomes the ____________________________________________________.

Example 1: Find the inverse of each relation.a) {(– 3, 26), (2, 11), (6, –1), (–1, 20)} b)

The graphs of relations can be used to find and graph inverse relations.

Example 2: Graph the inverse of each relation.

a) b)

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Inverse function: A linear relation that is described by a function has an inverse function that can ___________________________

of the ________________________________. The inverse of the function f (x) can be written as ____________ and is read f of x inverse or the inverse of f of x.

Example 3: Find the inverse of each function.

a) f ( x )=−3 x+27 b) f ( x )=54

x−8

Example 4: Carter sells paper supplies and makes a base salary of $2200 each month. He also earns 5% commission on his total sales. His total earnings f (x) for a month which he compiled x dollars in total sales is f ( x )=2200+0.05 x .

a) Find the inverse function.

b) What do x and f−1(x) represent in the context of the inverse function?

c) Find Carter’s total sales for last month if his earnings for that month were $3450.

Example 5: The table of values represents all points in the function f(x). Find the value of f-1(4).54

x f(x)

-5 0

-2 2

0 4

1 -5

4 34.7 Textbook Homework

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Find the inverse of each relation.

1. {(–2, 1), (–5, 0), (–8, –1), (–11, 2)} 2. {(3, 5), (4, 8), (5, 11), (6, 14)}

3. {(5, 11), (1, 6), (–3, 1), (–7, –4)} 4. {(0, 3), (2, 3), (4, 3), (6, 3)}

Graph the inverse of each function.

5. 6. 7.

Find the inverse of each function.

8. f (x) = 65

x – 3 9. f (x) = 4 x+2

3 10. f (x) =

3 x−16

11. f (x) = 3(3x + 4) 12. f (x) = –5(–x – 6) 13. f (x) = 2 x−3

7

Write the inverse of each equation in f−1(x) notation.

13. 4x + 6y = 24 14. –3y + 5x = 18 15. x + 5y = 12

16. 5x + 8y = 40 17. –4y – 3x = 15 + 2y 18. 2x – 3 = 4x + 5y

19. CHARITY Jenny is running in a charity event. One donor is paying an initial amount of $20.00 plus an extra $5.00 for every mile that Jenny runs.

a. Write a function D(x) for the total donation for x miles run.

b. Find the inverse function, D−1(x).

c. What do x and D−1(x) represent in the context of the inverse function?

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20. BUSINESS Alisha started a baking business. She spent $36 initially on supplies and can make 5 dozen brownies at a cost of $12. She charges her customers $10 per dozen brownies.

a. Write a function C(x) to represent Alisha’s total cost per dozen brownies.

b. Write a function E(x) to represent Alisha’s earnings per dozen brownies sold.

c. Find P(x) = E(x) – C(x). What does P(x) represent?

d. Find C−1(x), E−1(x), and P−1(x).

e. How many dozen brownies does Alisha need to sell in order to make a profit?

21. GEOMETRY The area of the base of a cylindrical water tank is 66 square feet. The volume of water in the tank is dependent on the height of the water h and is represented by the function V(h) = 66h. Find V −1 (h). What will the height of the water be whenthe volume reaches 2310 cubic feet?

22. SERVICE A technician is working on a furnace. He is paid $150 per visit plus $70 for every hour he works on the furnace.

a. Write a function C(x) to represent the total charge for every hour of work.

b. Find the inverse function, C−1(x).

c. How long did the technician work on the furnace if the total charge was $640?

23. FLOORING Kara is having baseboard installed in her basement. The total cost C(x) in dollars is given by C(x) = 125 + 16x, where x is the number of pieces of wood required for the installation.

a. Find the inverse function C−1(x).

b. If the total cost was $269 and each piece of wood was 12 feet long, how many total feet of wood were used?

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24. BOWLING Libby’s family went bowling during a holiday special. The special cost $40 for pizza, bowling shoes, and unlimited drinks. Each game cost $2. How many games did Libby bowl if the total cost was $112 and the six family members bowled an equal number of games?

Algebra 1 Name:

Review 4.1, 4.4, 4.7, 5.6 Period:

You must show ALL work for any credit.

1) Write an equation in slope-intercept form of the line with a slope of 2 and a y-intercept of –3. Then, graph the equation.

2) Write the equation 6x + 4y = 12 in slope-intercept form. Then, graph the equation.

m = _________ m = _________

b = _________ b = _________

3) Write an equation in point-slope form, slope-intercept form, and standard form for the line that passes through (2, 3) and is parallel to the line y = 3x + 1.

4) Write an equation in point-slope form, slope-intercept form, and standard form for the line that passes through (–4, –2) and is parallel to the line

y = 14 x – 1.

For 5 – 7, determine whether the graphs of each pair of equations are parallel, perpendicular, or neither. Show your work and explain.

5) – 3 x+4 y=8 6) 6 x−2 y=−2 7) y=25

x+2

– 4 x+3 y=– 6 y=3 x−4 5 x+2 y=4

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For 8 and 9, find the inverse of each relation. 8) {(2, –1), (5, –2), (6, 9), (7, 5)} 9)

For 10 and 11, find the inverse of each function. 10) f ( x )=6x+3 11) f (x)=4 (3 x – 5)

12) Andy is purchasing season tickets for the L.A. Lakers games. The ticket package requires a one-time purchase of a personal seat license costing $600. A ticket to each game costs $70. The cost C(x) in dollars for Andy for the first season is C ( x )=70 x+600, where x is the number of games Andy attends.

a) Find the inverse function

b) What do x and C-1(x) represent in the context of the inverse function?

c) How many games did Andy attend if his total cost for the season was $950?

For 13 – 15, graph each of the following inequalities. 13) 2 x+ y<3 14) 3 x−6 y ≥12 15) −3 x+2 y<6

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x y-2 10 34 -4




Algebra 1Section 5.3 Notes: Solving Multi-Step Inequalities

Multi-step inequalities can be solved by _________________________________________in the same way you would solve a multi-

step _________________________

Example 1:

a) Adriana has a budget of $115 for faxes. The fax service she uses charges $25 to activate an account and $0.08 per page to send faxes. How many pages can Adriana fax and stay within her budget? Use the inequality 25+0.08 p≤115. Graph the solution.

b) The Print Shop advertises a special to print 400 flyers for less than the competition. The price includes a $3.50 set-up fee. If the competition charges $35.50, what does the Print Shop charge for each flyer?

When multiplying or dividing by a negative number, the __________________ of the inequality symbol ____________________.

Example 2: Solve the inequality.

a) 23≥10−2 w b) 13−11d≥ 79 c) 43>−4 y+11

You can translate sentences into multi-step _________________________ and then solve them using the Properties of Inequalities.

Example 3: Define a variable, write an inequality, and solve the problem. Then check your solution.

a) Four times a number plus twelve is less than the number minus three.

b) Two more than half of a number is greater than twenty-seven.

When solving inequalities that contain ______________________________________, use the _______________________________

to remove the grouping symbols first. Then use the ______________________________________to simplify the resulting inequality.

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Example 4: Solve each inequality. Graph the solution on a number line.

a) 6 c+3 (2−c ) ≥−2 c+1 b) 6(5 z−3)≤ 36 z c) 2 (h+6 )>−3(8−h)

If solving an inequality results in a statement that is __________________, the solution set is the set of ________________________.

This solution set is written as ____________________________. If solving an inequality results in a statement that is ____________,

the solution set is the ______________________, which is written as the symbol ____. The empty set has no members.

Example 5: Solve each inequality. Check your solution.

a) −7 (k+4 )+11 k≥ 8k−2 (2 k+1 ) b) 2 (4 r+3 )≤ 22+8 (r−2 )

c) 18−3 (8c+4 )≥−6(4 c−1) d) 46 ≤ 8m−4(2m+5)

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Example 6: A student solved the inequality as shown below. Determine if their solution is correct. If not, determine in which step the mistake was made, and identify the mistake.

Given: −3 ( x−4 )+6 x−2 ≥ x−4Step 1: −3 x+12+6 x−2≥ x−4Step 2: 3 x+10≥ x−4Step 3: 4 x+10≥−4Step 4: 4 x≥−14

Step 5: x ≥−72

Example 7: Which statement is true for the inequality shown? 5 ( x−1 )+2 x>7 x−5A. The inequality is only true for numbers greater than 0B. The inequality is only true for numbers less than 0C. The inequality is never trueD. The inequality is true for all values of x.

Example 8: In the inequality 4 x+7≤ 451, x represents the number of T-shirts a printing company makes each day.Which statement MOST accurately describes how many T-shirts the company makes each day?

A. Less than 111 T-shirtsB. More than 111 T-shirtsC. Exactly 111 T-shirtsD. At most 111 T-shirts

Example 9:Cyndy wants to buy some new make-up, but she cannot afford more than $35 before the sales tax is added. The eye shadow she wants are priced $8 each and the eye liner she wants are priced $5 each. Which inequality could be used to determine s, the number of eye shadows, and l, the number of eye liners Cyndy can afford?

A. 8 s+51≥35B. 8 s+51 ≤35C. 5 s+81≥35D. 5 s+81 ≤35

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66


Justify each indicated step

1. 34

t – 3 ≥ –15 2. 5(k + 8) – 7 ≤ 23

34

t – 3 + 3 ≥ –15 +3 a. ? 5k + 40 – 7 ≤ 23 a. ?

34

t ≥ –12 5k + 33 ≤ 23

43

( 34 )t ≥

43

(–12) b. ? 5k + 33 – 33 ≤ 23 – 33 b. ?

t ≥ –16 5k ≤ –10

5 k5

≤ −10

5k ≤ –2

Solve each inequality. Check your solution.

3. –2b + 4 > –6 4. 3x + 15 ≤ 21 5. d2

– 1 ≥ 3

6. 55

a – 4 < 2 7. – t5

+ 7 > –4 8. 34

j – 10 ≥ 5

9. – 23

f + 3 < –9 10. 2p + 5 ≥ 3p – 10 11. 4k + 15 > –2k + 3

12. 2(–3m – 5) ≥ –28 13. –6(w + 1) < 2(w + 5) 14. 2(q – 3) + 6 ≤ –10

Define a variable, write an inequality, and solve each problem. Check your solution.

15. Four more than the quotient of a number and three is at least nine.

16. The sum of a number and fourteen is less than or equal to three times the number.

17. Negative three times a number increased by seven is less than negative eleven.

18. Five times a number decreased by eight is at most ten more than twice the number.

19. Seven more than five sixths of a number is more than negative three.

20. Four times the sum of a number and two increased by three is at least twenty-seven.

21. BEACHCOMBING Jay has lost his mother’s favorite necklace, so he will rent a metal detector to try to find it. A rental company charges a one-time rental fee of $15 plus $2 per hour to rent a metal detector. Jay has only $35 to spend. What is the maximum amount of time he can rent the metal detector?

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22. AGES Bobby, Billy, and Barry Smith are each one year apart in age. The sum of their ages is greater than the age of their father, who is 60. How old can the oldest brother can be?

23. TAXI FARE Jamal works in a city and sometimes takes a taxi to work. The taxicabs charge $1.50 for the first 15

mile and $0.25

for each additional 15

mile. Jamal has only $3.75 in his pocket. What is the maximum distance he can travel by taxi if he does not

tip the driver?

24. PLAYGROUND The perimeter of a rectangular playground must be no greater than 120 meters, because that is the total length of the materials available for the border. The width of the playground cannot exceed 22 meters. What are the possible lengths of the playground?

25. MEDICINE Clark’s Rule is a formula used to determine pediatric dosages of over-the-counter medicines.

weight of child( lb)150

× adult dose = child dose

a. If an adult dose of acetaminophen is 1000 milligrams and a child weighs no more than 90 pounds, what is the recommended child’s dose?

b. This label appears on a child’s cold medicine.What is the adult minimum dosage in milliliters?

Weight (lb) Age (yr) Dose

under 48 under 6 call a doctor

48-95 6-11 2 tsp or 10 mL

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c. What is the maximum adult dosage in milliliters?

5-4 PracticeSolving Compound InequalitiesGraph the solution set of each compound inequality.

1. –4 ≤ n ≤ 1 2. –4 ≤ p ≤ 4

3. x > 3 or x < 0 4. g < –3 or g ≥ 4

Write a compound inequality for each graph.

5. 6.

7. 8.

Solve each compound inequality. Then graph the solution set.

9. –n > 2 or 2n – 3 > 5 10. k – 3 < –7 or k + 5 ≥ 8

11. 5 < 3h + 2 ≤ 11 12. −14 < 3c + 1 < 13

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Algebra 1Section 5.5 Inequalities Involving Absolute Value

Warm-Up Solve each inequality.

1. a4<16 2.

57

p>−20

3. Define a variable and write an inequality for the phrase: One-half of Dan’s savings is less than $60.

Solving Absolute Value Inequalities with Less ThanWhen solving absolute value inequalities, there are two cases to consider.Case 1: The expression inside the absolute value symbols is ________________________.

Case 2: The expression inside the absolute value symbols is ___________________.**The solution is the intersection of the solutions of these two cases.

The inequality |x|<3 means that the distance between x and 0 is less than 3.

So, ____________________________________________________________.

Example 1: Solve the inequality. Then graph the solution set.

a) ¿n−3∨≤ 12 b) |x+6|←8 c) |x+6|<8

Example 2:

a) The average annual rainfall in California for the last 100 years is 23 inches. However, the annual rainfall can differ by 10 inches from the 100 year average. What is the range of annual rainfall for California?

b) A recent survey showed that 65% of young adults watched online video clips. The margin of error was within 3% points. Find the range of young adults that watch video clips.

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Solving Absolute Value Inequalities with Greater Than

Again, we must consider both cases.

Case 1: The expression inside the absolute value symbols is ________________________.

Case 2: The expression inside the absolute value symbols is _________________.

The inequality |x|>3 means that the distance between x and 0 is greater than 3.

So, _____________________________________________________________________________.

Example 3: Solve the inequality. Then graph the solution set.

a) |3 y−3|>9 b) |2 x+7|≥−11

c) |2 x+7|≥11

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75


Match each open sentence with the graph of its solution set.

1. |x−3| ≥ 1 a.

2. |2 x+1| < 5 b.

3. |5−x| ≥ 3 c.

Express each statement using an inequality involving absolute value.

4. The height of the plant must be within 2 inches of the standard 13-inch show size.

5. The majority of grades in Sean’s English class are within 4 points of 85.

Solve each inequality. Then graph the solution set.

6. |2z – 9| ≤ 1 7. |3 – 2r| > 7

8. |3t + 6| < 9 9. |2g – 5| ≥ 9

Write an open sentence involving absolute value for each graph.

10. 11.

12. 13.

14. RESTAURANTS The menu at Jeanne’s favorite restaurant states that the roasted chicken with vegetables entree typically contains 480 Calories. Based on the size of the chicken, the actual number of Calories in the entree can vary by as many as 40 Calories from this amount.

a. Write an absolute value inequality to represent the situation.

b. What is the range of the number of Calories in the chicken entree?

15. SPEEDOMETERS The government requires speedometers on cars sold in the United States to be accurate within ±2.5% of the actual speed of the car. If your speedometer reads 60 miles per hour while you are driving on a highway, what is the range of possible actual speeds at which your car could be traveling?

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16. BAKING Pete is making muffins for a bake sale. Before he starts baking, he goes online to research different muffin recipes. The recipes that he finds all specify baking temperatures between 350°F and 400°F, inclusive. Write an absolute value inequality to represent the possible temperatures t called for in the muffin recipes Pete is researching.

17. ARCHERY In an Olympic archery event, the center of the target is set exactly 130 centimeters off the ground. To get the highest score of ten points, an archer must shoot an arrow no further than 3.05 centimeters from the exact center of the target.

a. Write an absolute value inequality to represent the possible distances d from the ground an archer can hit the target and still score ten points.

b. Graph the solution set of the inequality you wrote in part a.

18. CATS During a recent visit to the veterinarian’s office, Mrs. Van Allen was informed that a healthy weight for her cat is approximately 10 pounds, plus or minus one pound. Write an absolute value inequality that represents unhealthy weights w for her cat.

19. STATISTICS The most familiar statistical measure is the arithmetic mean, or average. A second important statistical measure is the standard deviation, which is a measure of how far the individual scores deviate from the mean. For example, in a recent year the mean score on the mathematics section of the SAT test was 515 and the standard deviation was 114. This means that people within one deviation of the mean have SAT math scores that are no more than 114 points higher or 114 points lower than the mean.

a. Write an absolute value inequality to find the range of SAT mathematics test scores within one standard deviation of the mean.

b. What is the range of SAT mathematics test scores ±2 standard deviation from the mean?

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