EXAMPLE1 Translating Verbal Phrases · Verbal phrase Expression a. A number increased by 3 b. 9...

Lesson 7.1 Writing Expressions and Equations | 143

Goal: Write variable expressions and equations.

Writing Expressions and Equations

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Verbal phrase Expression

a. A number increased by 3

b. 9 less than a number

c. 1 more than three times a number

d. 5 decreased by the quotient of a number and 2

E X A M P L E 1 Translating Verbal Phrases

Vocabulary

Verbal model:

Verbal sentence Equation

a. 12 increased by a number is 18.

b. The difference of a number and 6 equals �2.

c. The product of �23

� and a number is 15.

d. �2 is equal to five times the sum of a number and 3.

E X A M P L E 2 Translating Verbal SentencesWhen translating

verbal sentences intoequations, look for

the key words "is" and"equals," which canbe represented by

the symbol =.

144 | Chapter 7 Notetaking Guide

Dinner The cost of a fish dinner is 3 times the cost of a chef salad. The fish dinner costs $21. Find the cost of the chef salad.

Solution

Write a verbal model.

Let s represent the cost of the salad.

3 times the cost of a � Cost of a

�

Use mental math: Because 3 times is 21, s � .

Answer: The cost of a chef salad is $ .

E X A M P L E 3 Writing and Solving an Equation

Assign ameaningful variable

to represent what youneed to find. InExample 3, s is

chosen to representthe price of a

salad.

Write the verbal phrase or sentence as a variable

expression or equation. Let n represent the number.

Guided Practice

1. 9 added to a number 2. �14

� of a number increased by 18

3. 24 divided by a number equals 6. 4. 26 minus 4 times a number is 10.

Use mental math to solve the following problem.Guided Practice

5. This year, the enrollment at the local junior college dropped by 500 to 4250. Write and solve an equation to find the enrollment last year.

Practice BFor use with pages 337–341

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Copyright © McDougal Littell/Houghton Mifflin CompanyAll rights reserved.

6 McDougal Littell Math, Course 2Chapter 7 Resource Book

Write the verbal phrase as a variable expression. Let x representthe number.

1. A number added to 10 2. 14 decreased by a number

3. 4 times a number 4. �13 increased by a number

5. 11 decreased by the quotient of 9 and a number

6. Twice a number subtracted from 1

Write the verbal sentence as an equation. Let y representthe number.

7. 15 increased by a number equals 27.

8. The difference of a number and 2 is 19.

9. The sum of twice a number and 7 is 32.

10. �13

� of a number decreased by 13 equals 45.

Write a verbal phrase for the variable expression.

11. x � 12 12. 9 � a 13. m � 5

Write a verbal sentence for the equation.

14. b � 3 � 7 15. 8y � 27 16. 11 � 2x � 30

Write the real-world phrase as a variable expression. Be sure toidentify what the variable represents.

17. 1 mile more than yesterday’s run 18. Two times your previous high score

19. One-third of the recipe 20. 3 inches shorter than your other dog

21. Yosemite National Park has many natural waterfalls within itsboundaries, including Horsetail Fall and Yosemite Falls. HorsetailFall, which is 1000 feet tall, is 1425 feet shorter than Yosemite Falls.Write an equation to find the height of Yosemite Falls. Then usemental math to solve the equation.

22. The population of Cape Coral, Florida increased by 27 thousandpeople from 1990 to 2000. In 2000, the population of Cape Coralwas 102 thousand people. Write an equation to find the populationof Cape Coral in 1990. Then use mental math to solve the equation.

23. In 2001, the cost of mailing a letter was 17 times the cost of mailinga letter in 1885. If it cost $.34 to mail a letter in 2001, find the costof mailing a letter in 1885.

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McDougal Littell Math, Course 2 7

Chapter 7 Resource Book

Practice CFor use with pages 337–341

Write the verbal phrase as a variable expression. Let x representthe number.

1. The sum of 8 and twice a number 2. �23

� of a number decreased by 14

3. �8 increased by the quotient of 13 and a number

Write the verbal sentence as an equation. Let y representthe number.

4. The difference of �2 and a number equals 15.

5. The sum of twice a number and 14 is �8.

6. �34

� of a number decreased by �2 is 24.

Write a verbal phrase for the variable expression.

7. 5 � y 8. 2b � 1 9. 5 � n � 3

Write a verbal sentence for the equation.

10. a � 6 � 14 11. 7 � 3x � 15 12. �12

�x � 8 � 32

Write the real-world phrase as a variable expression. Be sure toidentify what the variable represents.

13. 14 minutes longer than last week’s run 14. Three-quarters of the total time

15. 3 times higher than your previous score 16. 4�12

� inches shorter than your other dog

17. The population of Alexandria, Virginia increased by 17 thousandpeople from 1990 to 2000. In 2000, the population of Alexandriawas 128 thousand people. Write an equation to find the populationof Alexandria in 1990. Then use mental math to solve the equation.

18. In 2000, the number of visitors to Cuyahoga Valley National Parkin Ohio was about �

210� of the total number of visitors to all national

parks in the United States. If 3300 thousand people visited CuyahogaNational Park in 2000, find the total number of visitors to all nationalparks in the United States in 2000.

19. Describe a real-world situation that can be represented by theexpression x � 8.

20. You rent two movies from a video store for $6.95 each and g videogames for $2.95 each. You have $25 to spend on the movies and thevideo games. Write an equation that represents the situation.

Lesson 7.1

Lesson 7.2 Simplifying Expressions | 145

Goal: Simplify variable expressions.

Simplifying ExpressionsL ES S

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Vocabulary

Term:

Like terms:

Coefficient:

Constant term:

Equivalent variableexpressions:

Simplify the expression 8x � 2 � 4x.

8x � 2 � 4x � Write expression as a sum.

� 8x � Commutative property of addition

� Distributive property

� Simplify.

� Rewrite without parentheses.

E X A M P L E 1 Combining Like Terms

AfterExample 1, the

step of using thedistributive propertyin order to combinelike terms will not

be shown.


Simplify the expression 2(m � 1) � 6.

2(m � 1) � 6 � Distributive property

� Write as a sum.

� Combine like terms.

E X A M P L E 3 Simplifying an Expression

Identify the coefficients, constant term(s), and like terms

of the expression. Then simplify the expression.

Guided Practice

1. �2n � 4 � 3n 2. 10 � 6p � 5p � 4 3. 3l � 9 � l � 6

Identify the coefficients, constant terms, and like terms of the expression x � 7 � 3x � 1.

First, write the expression as a sum: x � � � � 3x � � �.Coefficient is . Coefficient is .

x � � � � 3x � � �

E X A M P L E 2 Coefficients, Constant Terms, Like Terms

and are like terms.

and are like terms.

Lesson 7.2 Simplifying Expressions | 147

Construction A rectangular skylight in an office building is 3 times as long as it is wide. Write and simplify an expression for the perimeter of the skylight in terms of the width w.

Solution

Because the skylight is 3 times as long as it is wide, its length is .

Perimeter � 2l � 2w Formula for perimeter of a rectangle

� 2� � � 2w Substitute for l.

� � 2w Multiply.


Answer: An expression for the perimeter of the skylight is .

E X A M P L E 4 Writing and Simplifying an Expression

Complete the following exercise.Guided Practice

4. A rectangle is 4 inches longer than it is wide. Write and simplify anexpression for the perimeter of the rectangle in terms of the width w.


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Tell whether the statement is true or false.

1. The coefficients of the expression 8 � 5x � 4 � 3x are 5 and 3.

2. The constants of the expression �4 � 11x � 9 � 8x are �4 and 9.

3. In the expression 3x � 9 � 3 � 10x, 3x and 3 are like terms.

4. The expressions 15x � 9 � 4x � 2 and 11x � 7 are equivalent.

Identify the coefficients, constant term(s), and like terms of the expression.

5. 8x � 9 � 3x 6. 17 � 2a � 5a � 1

7. 7m � 7 � 6m � 6 8. �10 � 15r � 22r � 8

Match the expression with an equivalent expression.

9. 5x � 4 � 3x � 9 A. 2x � 3

10. 5(x � 3) � 3x � 7 B. 2x � 5

11. �5x � 6 � 7x � 9 C. 2x � 8

Simplify the expression.

12. 18n � 13 � 5n 13. 4x � 6 � 9x � 1

14. �12a � 7 � 4a � 7 15. �6 � 14r � 12r � 3

16. 3(5 � 4b) � 2 17. 6(3 � 2z) � 11z � 4

18. A nut mixture contains peanuts, walnuts, and cashews. In themixture, the amount of peanuts is three times the amount ofcashews, and the amount of walnuts is four times the amount ofcashews. Let x represent the amount of cashews. Write and simplifyan expression for the total amount of nuts in the mixture.

19. A rectangular sheet of plywood is seven times longer than it is wide.Write and simplify an expression for the perimeter of the rectanglein terms of the width w.

20. A basketball player scored 8 points total during the first and secondquarters of a game. During the third quarter, she scored three timesas many points as she did in the fourth quarter. Let x represent thenumber of points the player scored in the fourth quarter. Write andsimplify an expression to represent the total number of points theplayer scored during the entire game.

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Identify the coefficients, constant term(s), and like terms of the expression.

1. 19 � 3x � 7x � 1 2. 10a � 10 � 8a � 8

3. �14 � 9m � 35m � 6 4. �22w � 14w � 13 � 22

5. 5(z � 6) � 4z 6. �2(b � 3) � 7

Write and simplify an expression for the perimeter of the rectangle.

7. 8. 9.

Simplify the expression.

10. �18s � 4 � 12s � 7 11. 9 � 23 � 17a � 41a

12. �12 � 13x � 52x � 12 13. 3( y � 6) � 12

14. �2(b � 7) � 13b � 8 15. 6(5 � 3t) � 17 � 5t

16. 4(w � z) � 2w � 3z � 6 17. �6(r � 2s) � 3(2r � s)

18. A nut mixture contains peanuts, walnuts, and cashews. In themixture, the amount of peanuts is four times the amount of cashews,and the amount of walnuts is 1�

13

� times the amount of cashews. Let xrepresent the amount of cashews. Write and simplify an expressionfor the total amount of nuts in the mixture.

19. A rectangular sheet of plywood is eight times longer than it is wide.Write and simplify an expression for the perimeter of the rectanglein terms of the width w.

20. A basketball player scored �12

� as many points in the second quarterthan she scored during the first quarter. During the third quarter, shescored three times as many points as she did in the fourth quarter.Write and simplify an expression to represent the total number ofpoints the player scored during the entire game.

Tell whether the two expressions are equivalent.

21. �9 � 10x � 12; 22. 18c � 9c � 12; 23. 7y2 � 4y;

6x � 3 � 4x 3(3c � 4) 11y2

7m

3m � 1

5a

7a

x � 9

x

Lesson 7.2

Goal: Solve addition and subtraction equations.

Solve x � 3 � �1.

x � 3 � �1 Write original equation.

from each side.

� Simplify.

✓ Check x � 3 � �1 Write original equation.

� Substitute for x.

✓ Solution checks.

Solving Addition andSubtraction Equations

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E X A M P L E 1 Solving an Addition Equation

Vocabulary

Inverse operations:

Equivalent equations:

Subtraction Property of Equality

Words Subtracting the same number from each side of an equation

produces an equation.

Algebra x � a � b x � a � a � b

Lesson 7.3 Solving Addition and Subtraction Equations | 149

Solve �4 � y � 9.

�4 � y � 9 Write original equation.

� to each side.

� Simplify.

✓ Check �4 � y � 9 Write original equation.

� Substitute for y.


E X A M P L E 2 Solving a Subtraction Equation

Solve 7 � 4.1 � b � 1.

7 � 4.1 � b � 1 Write original equation.

� 4.1 � Commutative property of addition


� from each side.

� Simplify.

✓ Check 7 � 4.1 � b � 1 Write original equation.

� Substitute for b.


E X A M P L E 3 Combining Like Terms

Addition Property of Equality

Words Adding the same number to each side of an equation produces

an equation.

Algebra x � a � b x � a � a � b

WATCH OUT!

You can add or subtractvertically or horizontallyto solve equations, butremember to performthe same operation on each side of theequation.


Business Travel Carol is out of the office for 8 hours meeting with a client.She spends 0.75 hour driving to the client’s office, and 1.25 hours drivingback from the client’s office. How long was Carol at the client’s office?

Solution

Write a verbal model. Let h represent the number of hours Carol spentat the client’s office.

� � �

� Write equation.


� from each side.

� Simplify.

Answer: Carol spent hours at the client’s office.

Time awayfrom office

E X A M P L E 4 Writing and Solving an Equation

Solve the equation. Check your solution.Guided Practice

1. t � 7 � 12 2. n � 8 � 0 3. 6 � y � 4

4. r � 12 � 15 5. p � (�3.6) � 4.9 6. 2.7 � s � 1.9 � 5.2



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Tell whether the given value of the variable is a solution of the equation.

1. x � 15 � 20; x � 5 2. a � 12 � 13; a � 1 3. 7 � m � �31; m � 24

Solve the equation. Check your answer.

4. y � 6 � 15 5. n � 23 � �14 6. 18 � r � 7

7. a � 12 � 28 8. z � 24 � �9 9. 20 � s � 35

10. 3.6 � m � 2.5 11. c � 2.1 � 6.7 12. 4.2 � x � 1.4 � 7.5

13. t � �15

� � �130� 14. �

67

� � a � 1 15. ��12

� � x � �23

� � ��56

�

16. Describe and correct the error in solving the equation 1.8 � a � �4.5. 1.8 � a � �4.5

1.8 � a � 1.8 � 4.5 � 1.8

a � 2.7

Write the verbal sentence as an equation. Then solve the equation.

17. The difference of a number b and 8 is �15.

18. 9 more than a number x is 24.

Write and solve an equation to find the unknown side length.

19. Perimeter: 12 ft 20. Perimeter: 11.3 mm 21. Perimeter: 12.3 in.

22. A paperback version of a book costs $17.10. This cost is $2.89 lessthan the cost of the hardcover version of the book. Write and solvean equation to find the hardcover cost of the book.

23. During a recent trip to the gym, you worked out with free weights androde a stationary bike. You used a stationary bike for 28 minutes ofthe 75 minutes you spent at the gym. Write and solve an equation tofind the number of minutes you spent working out with free weights.

24. The left- and right-hand margins on a sheet of paper are both 1.25 inches wide. The total width of the sheet of paper is 8.5 inches.Write and solve an equation to find the width of the text area thatlies between the margins.

3.9 in.

2.7 in.2.6 in.

?1.8 mm 1.3 mm

5 mm

?

3 ft

4 ft

?




Lesson 7.3Practice CFor use with pages 346–352

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Solve the equation. Check your answer.

1. y � 29 � 41 2. a � 15 � �8 3. 42 � m � �20.5

4. 3.6 � r � 2.3 5. �7.6 � q � 8.1 � 5 6. b � 12.4 � 9.8

7. 9.6 � w � 2.5 � �8 8. 6.2 � 3t � 4 � 2t 9. 6.8 � p � 9.5 � 7.3

10. 6x � 5x � �45

� � �27

� 11. n � �38

� � �23

� 12. �23

� � x � �15

�


13. The difference of a number n and 14 is �2.6.

14. 4.5 less than a number w is 1.

15. 5.3 more than a number b is �4. 16. The sum of 0.8 and a number z is 2.4.

17. A number y decreased by �12

� is �45

�. 18. �23

� increased by a number n is ��14

�.

Write and solve an equation to find the unknown side length.

19. Perimeter: 24 ft 20. Perimeter: 20 mm 21. Perimeter: 3.9 in.

22. A paperback version of a book costs $19.95. This cost is $1.61 lessthan the cost of the hardcover version of the book. Write and solvean equation to find the hardcover cost of the book.

23. During a recent trip to the gym, you worked out with free weightsand ran on the track. You ran on the track for 38 minutes of the 95 minutes you spent at the gym. Write and solve an equation to find the number of minutes you spent working out with free weights.

24. The left- and right-hand margins on a sheet of paper are both 0.75 inch wide. The total width of the sheet of paper is 8.5 inches.Write and solve an equation to find the width of the text area that lies between the margins.

25. Solve the equation �13

� � �257� � �

49

� � 8m � �23

� � 9m � �227�.

26. Write an addition equation and a subtraction equation that both havea solution of 5.

0.5 in.

1.2 in.

?

0.8 in.4.1 mm

8.2 mm

?

3.9 mm6 ft

8 ft

?

Lesson 7.4 Solving Multiplication and Division Equations | 151

Goal: Solve multiplication and division equations.

Solving Multiplication andDivision Equations

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Solve �30 � 6x.

�30 � 6x Write original equation.

each side by .

� Simplify.

✓ Check �30 � 6x Write original equation.



E X A M P L E 1 Solving a Multiplication Equation

Division Property of Equality

Words Dividing each side of an equation by the same nonzero number


Algebra ax � b (a 0) �aax� �

Multiplication Property of Equality

Words Multiplying each side of an equation by the same nonzero number


Algebra �ax

� � b (a 0) a p �ax

� �

�30 6x�

Solve �34

�x � �6.

�34

�x � �6 Write original equation.

� each side by .

� Simplify.

E X A M P L E 3 Solving an Equation Using a Reciprocal


Solve �4x

� � 0.3.

�4x

� � 0.3 Write original equation.

� each side by .

� Simplify.

E X A M P L E 2 Solving a Division Equation

In yournotebook, you

may want to compareand contrast solvingmulitiplication anddivision equations.This will help youremember how tosolve these types

of equations.

Need helpwith multiplying bya reciprocal? Seepage 237 of your

textbook.


1. 9v � 36 2. �8b � 96

3. �1.7 � �3k

� 4. �4d

� � 15

Rollerblading A woman is rollerblading through the park. You measure a 75-foot stretch of sidewalk, and count that she skates that portion of thesidewalk in 12 seconds. What is the speed of the woman?

A �625 feet per second B 3 feet per second

C 6.25 feet per second D 50 feet per second

Solution

Use the formula d � rt.

d � rt Write formula for distance.

� Substitute for d and for t.

� each side by .

� Simplify.

Answer: The speed of the woman is feet per second. The correct

answer is . A B C D

E X A M P L E 4 Standardized Test Practice

Lesson 7.4 Solving Multiplication and Division Equations | 153

Solve the following problem.Guided Practice

7. A filmmaker makes an edited version of his movie that is 120 minuteslong. The unedited footage is 7 times as long as the edited version. Write and solve an equation to find the length of the unedited film.

5. 6q � 4q � 16 6. �58

�m � 10

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1. �5r � 125; r � �15 2. 4.2a � �21; a � �5 3. ��

n6� � �84; n � �14

Describe how to solve the equation without actually solving.

4. 8x � 72 5. �14b � 8 6. �1m1� � �6

Solve the equation. Check your solution. Round the solution tothe nearest hundredth if necessary.

7. 4p � 48 8. 2.3y � �20.7 9. ��51

� c � 35

10. �9d � �76.5 11. �m7

� � �43 12. �6z.2� � 4.5

13. ��

a8� � 3.6 14. �9.8 � �

�

w2.3� 15. �6 � �

38

� r

16. �35

� t � 30 17. 5.3q � 1.431 18. b � 4b � 8


19. The quotient of a number x and 6 is 8.7.

20. Three times a number a equals 14.4.

21. The product of �2.2 and a number m is 13.2.

22. At a part-time job, Marcus earns $8.50 per hour. Write and solve an equation to find the number of hours he has to work to earn $102.

23. It costs you $1.96 for four pounds of bananas. Write and solve anequation to find the cost of one pound of bananas.

24. In a survey about favorite book categories, �14

� of the total number of people surveyed, or 39 people, responded that mystery was theirfavorite category. Write and solve an equation to find the totalnumber of people surveyed.

Lesson 7.4


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Describe how to solve the equation without actually solving.

1. 9x � 72 2. �1b3� � �9 3. �

83

� m � �3

Solve the equation. Check your solution. Round the solution tothe nearest hundredth if necessary.

4. 5.2y � �41.6 5. ��35

� c � 42 6. �8d � �349.6

7. �m7

� � �51 8. �4z.4� � 8.7 9. �

�

a6� � 3.5

10. �5.3 � ��

w7.1� 11. �8 � �

49

� r 12. �73

� t � 21

13. 2.6q � 1.872 14. b � 7b � 9 15. 14 � 7p � 9p


16. The quotient of a number y and 7 is �3.9.

17. �34

� of a number n is equal to �150.

18. The product of �1.7 and a number s is 13.6.

19. At a part-time job, Sheila works 22.5 hours a week. Write and solvean equation to find the amount of money she gets paid in one hour if she earns $207 for the 22.5 hours of work.

20. It costs you $4.50 for 2.5 pounds of cheese. Write and solve anequation to find the cost of one pound of cheese.

21. In a survey about favorite book categories, �23

� of the total number of people surveyed, or 64 people, responded that fiction was theirfavorite category. Write and solve an equation to find the totalnumber of people surveyed.

22. Which two equations have the same solution? Explain your reasoning.

A. �8x

� � 17 B. 17x � 8 C. 8x � �117� D. 0.125x � 17

Less

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Goal: Solve two-step equations.

Solve 2m � 7 � �19.

2m � 7 � �19 Write original equation.

� to each side.

� Simplify.

� each side by .

� Simplify.

Solving Two-Step EquationsL ES S

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E X A M P L E 1 Solving a Two-Step Equation

Solve �5p

� � 7 � �2.

�5p

� � 7 � �2 Write original equation.

� from each side.

� Simplify.

� each side by .

� Simplify.

E X A M P L E 2 Solving a Two-Step Equation

Don't forgetto check your solutionby substituting back

into the originalequation.

Lesson 7.5 Solving Two-Step Equations | 155

Long Distance Calls A long distance phone company charges customers a $5 monthly fee plus $3 per hour for long distance phone calls. One customer’s bill was $23. How many hours of long distance calls did the customer make?

Solution

Write a verbal model. Let h represent the number of hours of long distancethe customer used.

Monthly�

Hourly cost ofp

Hours of � Total costfee long distance long distance

� p �

� Write equation.

� from each side.

� Simplify.

� each side by .

� Simplify.

Answer: The customer made hours of long distance phone calls.

E X A M P L E 3 Writing and Solving a Two-Step Equation


1. 7q � 4 � 10 2. �6j� � 2 � 0 3. �

5y

� � 6 � �6

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1. 6x � 5 � 13; x � 3 2. 8m � 7 � �17; m � 3 3. 3c � 1 � �4; c � �1

Match the equation with its solution.

4. 4y � 3 � �1 A. y � �1

5. �3y � 4 � �1 B. y � �12

�

6. �4y � 3 � 1 C. y � 1

7. 3y � 4 � �1 D. y � 1�23

�

8. Put the steps for solving the equation 9x � 8 � �5 in order.

A. Divide each side by 9. B. Write original equation.

C. Check your answer. D. Add 8 to each side.

Solve the equation. Check your solution.

9. 7a � 4 � �17 10. �5s � 13 � �68 11. 12 � x � 19

12. �n6

� � 4 � 4 13. �3d.2� � 6 � 21 14. �

21

� p � 7 � �27

15. 0 � 14t � 26 16. �43

� m � 5 � 16 17. 3.4c � 1.7 � 6.8


18. Twice the number r increased by 15 equals �17.

19. 8 subtracted from 3 times a number c is 31.

20. A mail-order CD company is advertising a sale. During the sale,CDs are $6.95 each and the shipping and handling charge is only$5.25. How many CDs can you buy for $40?

21. A window is 21 inches wide, and its perimeter is 112 inches. What is the length of the window?

22. Kathy earns $445 a week for 40 hours of work and $25 an hour foreach hour over 40. How many hours did Kathy work if she earned$570 in one week?

Lesson 7.5


7.5

L ES S

ON Name Date



Describe the steps to solve the equation. Do not solve.

1. 11x � 7 � �41 2. ��21

� y � �38

� � �10

Solve the equation. Check your solution.

3. 8a � 5 � �27 4. �6s � 14 � �56 5. 38 � 7x � 3

6. �2n.5� � 7 � �82 7. �

5d.8� � 12 � �11.2 8. 14 � �

38

� p � �5

9. 0 � 18t � 32 10. ��23

� m � 7 � 19 11. 4.3c � 7.1 � 14.4

12. 4(x � 5) � �8 13. 3b � �134� � 4 14. �

49

� � 7x � ��59

�


15. �56

� of a number b increased by 18 equals �32.

16. 9 subtracted from 4 times a number t is 23.

17. A mail-order CD company is advertising a sale. During the sale,CDs are $7.95 each and the shipping and handling charge is only$4.35. How many CDs can you buy for $60?

18. A door is 36 inches wide, and its perimeter is 230 inches. What isthe length of the door?

19. Frank earns $475 a week for 40 hours of work and $28 an hour foreach hour over 40. How many hours did Frank work if he earned$601 in one week?

20. On the first three 100-point tests of the grading period, your scoreswere 88, 91, and 74. What score do you need to get on the fourth100-point test to have a mean of 86?

21. Write a two-step equation whose solution is 4.

22. Explain why you should check your answer after solving an equation.

Less

on 7

.5

Goal: Write and solve inequalities.

Inequality Verbal phrase Graph

a. x < 4 All numbers 4

b. x ≤ �2 All numbers

or �2

c. x > 3 All numbers

3

d. x ≥ �1 All numbers

or �10 1 2 3 4 5�5 �4 �3 �2 �1

0 1 2 3 4 5�5 �4 �3 �2 �1

0 1 2 3 4 5�5 �4 �3 �2 �1

0 1 2 3 4 5�5 �4 �3 �2 �1

Solving InequalitiesL ES S

O N


E X A M P L E 1 Graphing Inequalities

Vocabulary

Inequality:

Solution of an inequality:

Graph of an inequality:

Equivalent inequalities:

The inequalitysymbol ≤ is read

"is less than or equalto." The inequalitysymbol ≥ is read"is greater than or

equal to."

Lesson 7.6 Solving Inequalities | 157

Solve f � 3 ≤ �1. Then graph the solution.

f � 3 ≤ �1 Write original inequality.

to each side.

Simplify.

To graph , use a(n) dot and draw the arrow pointing

to the .

✓ Check To check the solution , choose any number

to substitute for f. Use f � 1 in the check below.

f � 3 ≤ �1 Write original inequality.

� 3 ≤?�1 Substitute for f.

≤ �1 Solution checks.

�2 �1 0 1 32 54 6

E X A M P L E 2 Solving an Inequality

Solve the inequality. Then graph the solution.Guided Practice

1. s � 1 ≥ 4 2. 4 < b � 3 3. w � 1 > �1

0 1�5 �4 �3 �2 �17654 8 9 105432 6 7 8


Solve �3m < �12. Then graph the solution.

�3m < �12 Write original inequality.

each side by . inequality.

Simplify.

To graph , use a(n) dot and draw the arrow pointing

to the .

6543210 7 8

E X A M P L E 3 Solving an Inequality

WATCH OUT!

Don’t forget to reversethe inequality when youmultiply or divide eachside of an inequality bya negative number.

Solve the inequality. Then graph the solution.Guided Practice

4. �4x

� ≥ 1 5. 5s < �30 6. �n ≥ �10

10987 11 12 13�5 �4 �3�6�7�8�94321 5 6 7




Lesson 7.6Practice BFor use with pages 366–370

7.6

L ES S

ON Name Date

Tell whether the given value of the variable is a solution of the inequality.

1. x � 7 ≤ �5; x � �8 2. 4a ≥ 36; a � 9 3. �3m < 4; m � ��32

�

Write an inequality represented by the graph.

4. 5.

6. 7.

8. Describe and correct the error in finding the solution to 8p ≥ �7.

Write an inequality to represent the situation. Then graph the inequality.

9. To ride an amusement park ride, you must be 48 inches tall or taller.

10. A restaurant can hold at most 40 people.

11. To run for the President of the United States, you must be at least 35 years old.

Solve the inequality. Then graph the solution.

12. x � 8 < 15 13. c � 12 ≥ �38 14. m � 24 ≤ �30

15. 14 � r > 43 16. �63 � w ≥ 120 17. �5a < 20

18. �8x

� ≥ �14

� 19. ��

n3� > 15 20. �

23

� q < �18

21. A wheelbarrow can carry at most 400 pounds. Write and solve aninequality to find the greatest number of 50-pound bags of concretethat the wheelbarrow can carry.

22. A book store sells used paperbacks for $3.75 each. You receive a $2 discount if you spend at least $30 in the store. Write and solve an inequality that represents the least number of paperbacks youmust buy in order to receive the discount.

23. You are mailing a 42-pound item by parcel post. The total weight of an item and its packaging cannot be greater than 70 pounds. Writeand solve an inequality that represents the heaviest the packagingcan be without exceeding the 70-pound weight limit.

�2 �1�3�4 0 1 2 3 4�3�6�9 0 3 6 9 12 15

�1�2�3�4�5�6 0 1 2�1 0 1 2 3 4 5 6 7

8p ≥ �7

�88p� ≤ �

�87�

p ≤ ��78

�



Less

on 7

.67.6

L ES S

ON Name Date


Tell whether the given value of the variable is a solution of the inequality.

1. �13

� x ≤ 4; x � 12 2. �4a < 8; a � ��32

� 3. ��34

� m ≤ ��14

�; m � �2

Write an inequality represented by the graph.

4. 5.

6. 7.

Write an inequality to represent the situation. Then graph the inequality.

8. To ride an amusement park ride, you must be 52 inches tall or taller.

9. A restaurant can hold at most 45 people.

10. In most states, you must be at least 16 years old to drive a motor vehicle.

Solve the inequality. Then graph the solution.

11. 17 � r > 51 12. �65 � w ≥ 134 13. �8x

� ≥ �34

�

14. �8x

� > �8 15. ��

n6� ≥ 12 16. �

25

� q ≤ �18

17. �11a < 44 18. 3x � 2 < 5 19. 2 � 7x ≤ 16

20. A wheelbarrow can carry at most 500 pounds. Write and solve aninequality to find the greatest number of 20-pound bags of concretethat the wheelbarrow can carry.

21. A book store sells used paperbacks for $3.75 each. You receive a $2 discount if you spend at least $36 in the store. Write and solve an inequality that represents the least number of paperbacks youmust buy in order to receive the discount.

22. You are mailing a 25-pound item and a 37-pound item in the samepackage by parcel post. The total weight of the items and theirpackaging cannot be greater than 70 pounds. Write and solve aninequality that represents the heaviest the packaging can be withoutexceeding the 70-pound weight limit.

23. Write a two-step inequality whose solution is x < 3.

�9 �8 �7 �6 �5 �4 �3 �2 �1�10 �5�15 0 5 10 15 2520

�2 �1�3 0 1 2 3�5 �40 1 2 3 4 5 6 7 8

Lesson 7.7 Functions and Equations | 159

Goal: Write and evaluate function rules.

Functions and EquationsL ES S

O N

Evaluate the function y � 3x when x � 8.

y � 3x Write rule for function.


� Multiply.

E X A M P L E 1 Evaluating a Function

Vocabulary

Function:

Input:

Output:

Domain:

Range:


Write a function rule for the input-output table.

Solution

You can see that you obtain each output by the input.

Answer: The function rule given by the table is .

E X A M P L E 3 Writing a Function Rule

Input x �2 �1 0 1 2 3 4

Output y �3.5 �2.5 �1.5 �0.5 0.5 1.5 2.5

Make an input-output table for the function y � x � 4.2 using thedomain 0, 1, 2, and 3. Then state the range of the function.

Solution

The range of the function is the set of outputs: , , ,

and .

E X A M P L E 2 Making an Input-Output Table

Input x 0 1 2 3

Substitution y � � 4.2 y � � 4.2 y � � 4.2 y � � 4.2

Output y

Complete the following exercise.Guided Practice

1. Make an input-output table for the function y � 4 � x using the domain�2, �1, 0, 1, and 2. Then state the range of the function.

Lesson 7.7 Functions and Equations | 161

Squares In the diagram of the squares, the input s is the length of each side of a square. The output P is the perimeter of the square. Write a rule for thefunction. Then use the rule to find the perimeter of a square with sides 9 units.

Solution

1. Begin by making an input-output table.

2. Notice that each output value is the input value. So, a rule

for the function is .

3. To find the perimeter of a square with sides 9 units, evaluate the

function when s � 9. Because P � � , the perimeter

of the square is .

1 unit2 units

3 units4 units

E X A M P L E 4 Writing a Function Rule From a Pattern

Input

Output

Write a function rule for the input-output table.Guided Practice

2. 3.

Input x �1 0 1 2

Output y �2 0 2 4

Input x 2 4 6 8

Output y �12

� 1 1�12

� 2


7.7

L ES S

ON Name Date



Evaluate the function y � 5x � 3 for the given value of x.

1. 7 2. 0 3. �4

Match the function with its possible range.

4. y � 2x � 1 A. �7, �4, �1, 2, 5, 8

5. y � �x � 2 B. �5, �3, �1, 1, 3

6. y � 3x � 1 C. 11, 6, 1, �4, �9, �14

7. y � �5x � 1 D. 4, 3, 2, 1, 0

Make an input-output table for the function using the domain �2, �1, 0, 1, and 2. Then state the range of the function.

8. y � x � 8 9. y � �15x 10. y � 4.3x

11. y � x � 2.75 12. y � 0.4x � 3 13. y � 18 � 3x


14. 15.

16. 17.

18. A custom case company makes travel cases for computer equipment. There is a 2-inch foam lining around the inside of each case. The function y � x � 4, where x is the width of a laptop computer, can be used to find the total width of a laptop case, including the foam lining. Create an input-output table using the domain 12, 15, 18, and 21.

19. A magazine costs $3.95 per issue. Write a function rule that models the cost y of x issues. Then use the function to calculate the cost of 6 issues.

20. It costs $.15 to make one copy on the copier at the local library.Write a function rule that models the cost y of making x copies.Then use the function to calculate the cost of 24 copies.

computer

x in.2 in. 2 in.

foam

Less

on 7

.7

Input x �3 �2 �1 0

Output y �12 �8 �4 0

Input x 0 1 2 3

Output y 1.2 2.2 3.2 4.2

Input x 0 1 2 3

Output y 5 4 3 2

Input x 0 1 2 3

Output y 3 4 5 6

7.7

L ES S

ON Name Date





Evaluate the function y � 5.4x � 2 for the given value of x.

1. 8 2. 0 3. �7

Make an input-output table for the function using the domain �2, �1, 0, 1, and 2. Then state the range of the function.

4. y � x � 13 5. y � �23x 6. y � 7.4x

7. y � x � 9.85 8. y � 1.2x � 2 9. y � 21 � 4x


10. 11.

12. 13.

Write a function rule for the points in the coordinate plane.

14. 15. 16.

17. A magazine costs $4.95 per issue. Write a function rule that modelsthe cost y of x issues. Then use the function to calculate the cost of12 issues.

18. A custom case company makes travel cases for computer equipment. There is a 2.5-inch foam lining around the inside of each case. Write a function rule that represents the total width y of the case in terms of the width x of a laptop computer. Then find the width of a case for a 15-inch wide laptop.

19. Write two different functions for which an input of �5 gives an output of 8.

computer

x in.2.5 in. 2.5 in.

foam

x

y

1 2 3 4�2�3�4

1

�2

�3

�4

2

3

4

O

x

y

1 2 3 4�2�3�4

1

2

3

4

5

6

7

8

O

x

y

1 2 3 4�2�3�4

1

�2

2

3

4

5

6

O

Lesson 7.7

Input x 4 5 6 7

Output y 44 55 66 77

Input x 0 1 2 3

Output y 1.6 2.6 3.6 4.6

Input x 2 3 4 5

Output y �0.2 �1.2 �2.2 �3.2

Input x �1 0 1 2

Output y �7 �6 �5 �4

Goal: Graph functions in a coordinate plane.

Graph the function y � 3x � 1.

1. Make an input-output table by choosing several input values and evaluating the function for the output values.

2. Use the table to write a list of ordered pairs:

3. Plot the ordered pairs in a coordinate plane.

4. Notice that all of the points lie on a line. Any other ordered pairs satisfying y � 3x � 1would also lie on the line whengraphed. The line represents the complete graph of the function y � 3x � 1.

x

y

1 2 3 4 5 6 7�2�3�4�5�6�7

1

�2

�3

�4

�5

�6

�7

2

3

4

5

6

7

O

Graphing FunctionsL ES S

O N


E X A M P L E 1 Graphing a Function

Vocabulary

Linear function:

When thedomain of a functionis not given, assumethat it includes everyx-value for which thefunction can produce

a correspondingy-value.

x Substitution y

�2

�1

0

1

2

Lesson 7.8 Graphing Functions | 163

The cost of gasoline is $1.50 per gallon. Write and graph a function thatrepresents the cost y of x gallons of gasoline.

The situation can be represented by the function , where y isthe total cost of x gallons of gasoline.

1. Make an input-output table. 2. Plot the ordered pairs and connect them.

x

y

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

O

E X A M P L E 2 Writing and Graphing a Function

Input x Output y

0

1

2

3

4

WATCH OUT!

In Example 2, note that you cannot haveless than 0 gallons ofgasoline, so you cannotuse any numbers lessthan 0 in the domain.

Graph the function.Guided Practice

1. y � x � 2 2. y � �2x

� 3. y � 4x � 1

x

y

1 2 3 4�2�3�4

1

�2�3�4

234

Ox

y

1 2 3 4�2�3�4

1

�2�3�4

234

Ox

y

1 2 3 4�2�3�4

1

�2�3�4

234

O


Tell whether each graph represents a function of x. If it does, tell whetherthe function is linear.

a. b. c.

Solution

a.

b.

c.

x

y

1 2 3 4�3�4

1

�2�3�4

234

Ox

y

1 2 3 4�2�3�4

�2�3�4

234

Ox

y

1 2 3 4�2�3�4

1

�2�3�4

234

O

E X A M P L E 3 Identifying Linear Functions

Recallthat a function

pairs each inputvalue with exactly

one outputvalue.




Lesson 7.8Practice BFor use with pages 376–385

7.8

L ES S

ON Name Date

Identify the graph of the function on the coordinate plane.

1. y � �5x

2. y � 4x

3. y � ��12

� x

Graph the function.

4. y � 3x 5. y � 8 � x 6. y � �14

� x

7. y � x � 5 8. y � 3x � 6 9. y � �2x � 3

10. y � �13

� x � 2 11. y � 0.75x � 3 12. y � �x � 8

Write and graph a function that converts the units.

13. x feet to y inches 14. x pounds to y ounces 15. x months to y years

Tell whether the graph represents a function of x. If it does,tell whether the function is linear or nonlinear.

16. 17. 18.

19. When you are swimming, your body burns about 8 calories everyminute. Write and graph a function that models the number of calories burned y after swimming for x minutes.

20. Outdoor carpeting costs $1.25 for each square foot. Write and graph a function that models the cost y of x square feet of carpeting.

21. A phone company charges a $.25 dialing fee for calling a long-distance number and then charges $.10 for each minute of the call. This situation can be represented by the function y � 0.1x � 0.25, where y is the total cost of the call and x isthe length of the call in minutes. Graph the function.

x

y

1 2 3 4 5 6

�2

�3

�4

2

3

4

O

x

y

1 2 3 4�2�3�4

1

�2

�3

2

5

O

x

y

1 2 3 4�2�3�4

�2

1

3

4

5

6

O

x

y

2 3 4�2�3�4

1

4

O



Less

on 7

.87.8

L ES S

ON Name Date


Graph the function.

1. y � 4x 2. y � 9 � x 3. y � �34

� x

4. y � 2x � 5 5. y � �6x � 1 6. y � �13

� x � 4

7. y � 0.25x � 1.5 8. y � �x � 11 9. 4x � y � 3

Write and graph a function that converts the units.

10. x inches to y feet 11. x quarts to y cups 12. x years to y months

Tell whether the graph represents a function of x. If it does,tell whether the function is linear or nonlinear.

13. 14. 15.

Graph the functions in the same coordinate plane. Then tell wherethey intersect.

16. y � x and y � 2x � 1 17. y � 2x � 3 and y � 9 � 4x

18. y � �x � 4 and y � 5x � 8

19. When you are jogging, your body burns about 10 calories everyminute. Write and graph a function that models the number of calories burned y after jogging for x minutes.

20. Outdoor carpeting costs $1.75 for each square foot. Write and graph a function that models the cost y of x square feet of carpeting.

21. A phone company charges a $.50 dialing fee for calling a long-distance number and then charges $.12 for each minute of the call. This situation can be represented by the function y � 0.12x � 0.5, where y is the total cost of the call and x isthe length of the call in minutes. Graph the function.

22. Graph the functions y � 2x and y � �x � 3 in the same coordinate plane and tell where they intersect. Then solve the equation 2x � �x � 3. What do you notice?

x

y

1 2 4 5 6�2

1

�2

�3

�4

2

3

4

Ox

y

1 2 3 4�2�3�4

1

�2

�3

�4

2

3

Ox

y

1 2 3 4�2�3�4

1

�4

2

3

4

O

EXAMPLE1 Translating Verbal Phrases · Verbal phrase Expression a. A number increased by 3 b. 9...

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