Worked-out Problems

Chapter 3 

Chapter Opener 

Math in the Real World (p. 117) 

0 10 20 30 40 50 60 70 

Prerequisite Skills Quiz (p. 118) 

1. Sample answer: An equation is a mathematical sentence 

formed by setting two expressions equal. 

Example: x 3 5 

2. 9(x 4) 9(x) 9(4) 9x 36 

3. 8(z 7) 8(z) 8(7) 8z 56 

4. 6(m 12) 6(m) (6)(12) 

6m (72) 

6m 72 

5. 10(n 5) 10(n) (10)(5) 

10n (50) 

10n 50 

6. c 4 c c c 4 0 4 4 

7. 9b 12b 3 3b 3 

8. 4(a 2) a 4a 8 a 4a a 8 5a 8 

9. 2(2d 5 d) 4d 10 2d 

4d 2d 10 

6d 10 

10. x 13 7 

x 13 13 7 13 

x 6 

Check: x 13 7 

6 13 7 

7 7 ✓ 

11. h 6 8 

6 p h 6 6(8) 

Check: 

h 48 

h 6 8 

48 8 

6 

8 8 ✓ 

12. q 9.6 2 

q 9.6 9.6 2 9.6 

q 11.6 

Check: q 9.6 2 

11.6 9.6 2 

2 2 ✓ 

13. 65 13b 

65 

13b 

 

13 

13 

5 b 

Check: 65 13b 

65 13(5) 

65 65 ✓ 

Lesson 3.1 

3.1 Concept Activity (p. 119) 

1. 1 2x 9 

 

 

 

The solution is 4. 

2. 4x 1 5 


3. 2x 2 8 

 

 

 


4. 9 2x 5 

 

 

 

 

 

 

Copyright © McDougal Littell/Houghton Mifflin Company 

All rights reserved. 


Chapter 3 

Pre-Algebra 67 

Worked-Out Solution Key

Chapter 3 continued 

5. 11 2 3x 


6. 5x 3 8. 

 

 

 

 

3. 1 2r 9 4. 2 6h 20 

1 9 2r 9 9 2 20 6h 20 20 

8 2r 

18 6h 

8 

2 r 

18 6h 

 

2 2 

6 6 

4 r 

3 h 

Check: 1 2r 9 Check: 2 6h 20 

1 2(4) 9 2 6(3) 20 

1 1 ✓ 2 2 ✓ 

5. To solve 3x 7 5, add 7 to each side of the 

equation. To solve 3x 7 5, subtract 7 from each 

side of the equation. 

6. b 4 8 1 7. c 2 6 

6 

b 4 8 8 1 8 c 2 2 6 2 

6 

 

 

b 4 9 c 8 

6 

4 b 4 4(9) 6 c 

6 6(8) 

b 36 c 48 

Check: b 4 8 1 Check: c 2 6 

6 

3 6 

8 

4 

1 4 8 

2 

6 

6 


7. The subtraction property of equality is used in Step 2. 

The division property of equality is used in Step 3. 

8. Step 1: 3x 6 12 

Step 2: 3x 6 

Step 3: x 2 

9. Sample answer: Subtract 1 from each side. 

2x 1 1 5 1 

2x 4 

Then divide each side by 2. 

2 x 

4 2 2 

x 2 

3.1 Checkpoint (pp. 120–121) 

1. 4x 1 5 2. 3n 8 2 

4x 1 1 5 1 3n 8 8 2 8 

4x 4 

3n 6 

4 x 

4 4 4 3 n 

6 

 

3 3 

x 1 

n 2 

Check: 4x 1 5 Check: 3n 8 2 

4(1) 1 5 3(2) 8 2 

5 5 ✓ 2 2 ✓ 

68 Pre-Algebra 

Chapter 3 Worked-Out Solution Key 

1 1 ✓ 6 6 ✓ 

8. 2 d 5 1 9. 12 f 8 

2 

2 1 d 5 1 1 12 8 f 8 8 

2 

3 d 5 20 f 

2 

5(3) 5 d 5 2(20) 2 

f 

2 

15 d 40 f 

Check: 2 d 5 1 Check: 12 f 8 

2 

2 1 5 

1 12 

5 

4 0 

8 

2 

2 2 ✓ 12 12 ✓ 

10. 12 4s 12 

12 4s 12 12 12 

4s 24 

4 

 

s 

4 

24 

 

4 

s 6 

Check: 12 4s 12 

12 4(6) 12 

1212 ✓ 


All rights reserved.


11. 6 2m 8 12. 2 5 n 

6 2m 6 8 6 2 5 5 n 5 

2m 2 

7 n 

2m 

2 

 

7 

 

n 

2 

2 

1 

1 

m 1 

7 n 

Check: 6 2m 8 Check: 2 5 n 

6 2(1) 8 2 5 7 

8 8 ✓ 2 2 ✓ 

3.1 Guided Practice (p. 122) 

1. You can use two inverse operations to solve a 

two-step equation. 

2. Sample answer: To solve 9 2s 15, subtract 9 from 

each side. 

9 2s 9 15 9 

2s 6 

Then divide each side by 2. 

2 s 

6 2 2 

s 3 

3. 5c 6 31 

5c 6 6 31 6 

5c 25 

5 c 

2 5 

 

5 5 

c 5 

Check: 5c 6 31 

5(5) 6 31 

31 31 ✓ 

4. 2 3 

t 11 

2 11 3 

t 11 11 

9 3 

t 

3(9) 3 3 

t 

27 t 

Check: 2 3 

t 11 

2 2 7 

11 

3 

2 2 ✓ 

5. 9z 4 5 

9z 4 4 5 4 

9z 9 

9 

 

z 

9 

9 

 

9 

z 1 

Check: 9z 4 5 

9(1) 4 5 

5 5 ✓ 

6. 8 8d 64 

8 8d 8 64 8 

8d 72 

8d 

72 

 

8 

8 

d 9 

Check: 8 8d 64 

8 8(9) 64 

64 64 ✓ 

7. (1) Total cost Cost for Cost for each Number 

p 

for repairs parts hour of labor of hours 

(2) 168 78 45h 

(3) 168 78 78 45h 78 

90 45h 

9 0 

4 5h 

 

45 

45 

2 h 

It took 2 hours to repair the car. 

3.1 Practice and Problem Solving (pp. 123–124) 

8. 12k 7 31 

12k 7 7 31 7 

12k 24 

1 2k 

2 4 

 

12 

12 

k 2 

Check: 12k 7 31 

12(2) 7 31 

31 31 ✓ 

9. 13n 42 81 

13n 42 42 81 42 

13n 39 

1 3n 

3 9 

 

13 

13 

n 3 

Check: 13n 42 81 

13(3) 42 81 

81 81 ✓ 



Chapter 3 




10. 56 17p 29 

56 29 17p 29 29 

85 17p 

8 5 

1 7p 

 

17 

17 

5 p 

Check: 56 17p 29 

56 17(5) 29 

56 56 ✓ 

11. w 21 3 

4 

w 21 21 3 21 

4 

w 4 18 

4 w 4 4(18) 

w 72 

Check: w 21 3 

4 

7 2 

21 

4 

3 

3 3 ✓ 

12. h 19 10 

9 

h 19 19 10 19 

9 

Check: 

h 9 9 

9 h 9 9(9) 

h 81 

h 19 10 

9 

8 1 

19 

9 

10 

10 10 ✓ 

d 

13. 25 29 

1 2 

d 

25 25 29 25 

1 2 

d 

4 1 2 

12 

d 

1 2 12(4) 

d 48 

d 

Check: 25 29 

1 2 

4 8 

25 

12 

29 

29 29 ✓ 



a 

14. 12 17 3 6 

a 

12 17 17 17 

3 6 

a 

5 

3 6 

36(5) 36 

a 

3 6 

180 a 

a 

Check: 12 17 3 6 

12 180 

17 

36 

12 12 ✓ 

15. 18 r 42 

18 r 18 42 18 

r 24 

r 

24 

 

1 1 

r 24 

Check: 18 r 42 

18 (24) 42 

42 42 ✓ 

16. 80 23 3v 

80 23 23 3v 23 

57 3v 

57 

3v 

3 3 

19 v 

Check: 80 23 3v 

80 23 3(19) 

80 80 ✓ 

17. 2q 63 47 

2q 63 63 47 63 

2q 110 

2 

 

q 

2 

1 10 

 

2 

q 55 

Check: 2q 63 47 

2(55) 63 47 

47 47 ✓ 




18. 2 

x 4 12 

2 

x 4 4 12 4 

2 

x 8 

2 2 

x 2(8) 

x 16 

Check: 2 

x 4 12 

( 16) 

4 

2 

12 

12 12 ✓ 

19. 5 19 7 

x 

5 19 19 7 

x 19 

14 7 

x 

7(14) 7 7 

x 

98 x 

Check: 5 19 7 

x 

5 19 ( 98) 

 

7 

5 5 ✓ 

20. a. Given: You begin with 16 gallons of gasoline. You 

use 3 gallons of gasoline per hour of driving. You will 

stop when there is 1 gallon of gasoline left. 

Find: When will you stop to refuel? 

b. Sample answer: Let h the number of hours. 

Total 

gallons 

Gallons Number 

p 

per hour of hours 

16 3h 1 

c. 16 1 3h 1 1 

15 3h 

1 5 

3 h 

 

3 3 

Gallons 

left 

5 h 

You will need to stop and refuel after 5 hours. 

Hours of Gallons of 

driving gasoline left 

0 16 

1 13 

2 10 

3 7 

4 4 

5 1 



21. Let x the discount per person. 

Price with Price without Number 

p 

discount discount of people 

729 810 9x 

729 810 810 9x 810 

81 9x 

81 

 

9 

9x 

 

9 

9 x 

The discount per person is $9. 

22. Let x the weight of one car. 

Total Weight of Number 

p 

weight locomotive of cars 

4725 125 50x 

4725 125 125 50x 125 

4600 50x 

46 00 

5 0x 

 

50 

50 

92 x 

Each car weighs 92 tons. 

23. 5 2x 7 

5 2x 5 7 5 

2x 2 

2x 

2 

 

2 

2 

x 1 

24. 32 9x 140 

32 9x 32 140 32 

9x 108 

9 

 

x 

9 

1 08 

 

9 

x 12 

25. 13 6x 67 

13 6x 13 67 13 

6x 54 

6 x 

5 4 

 

6 6 

x 9 

26. 8 3x 19 

8 3x 8 19 8 

3x 27 

3x 

27 

 

3 

3 

x 9 

Chapter 3 

Discount 

per person 

Weight of 

one car 




27. a. Let x the number of flocks of chicks. 

Total Cost of Cost per flock Flocks of 

p 

amount heifer of chicks chicks 

755 500 20x 

755 500 500 20x 500 

255 20x 

2 55 

2 0x 

 

20 

20 

12.75 x 

So, your class can buy 12 flocks of chicks. 

b. Let x the number of pigs. 

Total Cost of Cost per Number 

p 

amount heifer pig of pigs 

755 500 120x 

755 500 500 120x 500 

255 120x 

2 55 

1 20x 

 

120 

120 

2.125 x 

So, your class can buy 2 pigs. 

c. No. Sample answer: The cost of 1 heifer and 

2 pigs is $500 2($120) 740. So, there is 

$755 $740 $15. This is not enough to buy 

a flock of chicks. 

28. 54.7 9.3n 8.2 

54.7 8.2 9.3n 8.2 8.2 

46.5 9.3n 

46. 

5 

9.3n 

 

9. 

3 9.3 

5 n 

Check: 54.7 9.3n 8.2 

54.7 9.3(5) 8.2 

54.7 54.7 ✓ 

29. 5.7 2.6d 14.02 

5.7 2.6d 5.7 14.02 5.7 

2.6d 8.32 

2 .6d 

8. 

32 

 

2.6 

2. 

6 

d 3.2 

Check: 5.7 2.6d 14.02 

5.7 2.6(3.2) 14.02 

14.02 14.02 ✓ 



30. 3.2r 14.7 6.74 

3.2r 14.7 14.7 6.74 14.7 

3.2r 21.44 

3 . 2 

3. 

r 

2 

2 1.44 

3 

.2 

r 6.7 

Check: 3.2r 14.7 6.74 

3.2(6.7) 14.7 6.74 

6.74 6.74 ✓ 

31. 9.1 3 

k 

.7 4.1 

k 

9.1 4.1 4.1 4.1 

3 .7 

k 

5 3 .7 

3.7(5) 3.7 3 

k 

.7 

18.5 k 

k 

Check: 9.1 3 .7 4.1 

9.1 1 8. 

5 

4.1 

3. 

7 

9.1 9.1 ✓ 

32. 11.3 2 

p 

.8 1.5 

p 

11.3 11.3 1.5 11.3 

2 .8 

p 

2 .8 9.8 

2.8 2 

p 

.8 2.8(9.8) 

p 27.44 

p 

Check: 11.3 2 .8 1.5 

11.3 27 . 44 

 

2. 

8 

1.5 

1.5 1.5 ✓ 

33. 6.8 1 

c 

.2 2.9 

c 

6.8 6.8 2.9 6.8 

1 .2 

c 

1 .2 3.9 

1.2 1 

c 

.2 1.2(3.9) 

c 4.68 

c 

Check: 6.8 1 .2 2.9 

6.8 ( 4. 

68) 

 

1. 

2 

2.9 

2.92.9 ✓ 




34. Your friend’s method: 

18 2x 36 

18 2x 2x 36 2x 

18 36 2x 

18 36 36 2x 36 

54 2x 

5 4 

2 x 

 

2 2 

27 x 

Your method: 

18 2x 36 

18 2x 18 36 18 

2x 54 

2 

 

x 

2 

54 

 

2 

x 27 

Sample answer: Both methods produce the same answer. 

Your method involved fewer steps by leaving the variable 

on the left side of the equation. 

35. a. Let m the number of months. 

b. 

Amount 

already 

saved 

Amount saved ($) 

800 

600 

400 

200 

 

278 50m 

Amount 

saved per 

month 

p 

Saving for Class Trip 

y 

0 

0 2 4 6 8 10 12m 

Number of months 

Number 

of 

months 

Number of Amount of 

months from now money saved 

0 $278 

1 $328 

2 $378 

3 $428 

4 $478 

Sample answer: The points lie along a straight line. 

Continue plotting points until you obtain a y-value of 

850. The corresponding x-value will give the number 

of months it will take to save enough money for the 

trip. 

––CONTINUED–– 

35. ––CONTINUED–– 

c. Let m the number of months. 

Amount Amount Number 

Total 

already saved per p of 

amount 

saved month months 

850 278 50m 

850 278 278 50m 278 

572 50m 

5 72 

5 0m 

 

50 

50 

11.44 m 

It will take about 12 months to save for the trip. 

d. Sample Answer: 

Part (a) method: An advantage is that the table 

shows the exact amount saved for every month. A 

disadvantage is that a table may need to become very 

large in order to find a solution. 

Part (b) method: An advantage is that the scatter 

plot shows the change from month to month. A 

disadvantage is that you may need to plot a large 

number of points in order to find a solution. 

Part (c) method: An advantage is that an equation will 

give an exact solution. A disadvantage is that it does 

not show the amount saved for each month. 

36. To solve x 2 

2, first multiply each side by 4. 

4 

4 x 2 

 

4 4(2) 

x 2 8 

Then subtract 2 from each side. 

x 2 2 8 2 

x 6 

Check: x 2 

2 

4 

6 2 

 

4 

2 

8 4 2 

2 2 ✓ 

3.1 Mixed Review (p. 124) 

37. 11(6z 14) 11(6z) 11(14) 66z 154 

38. 9(2x 12) 9(2x) (9)(12) 

18x (108) 

18x 108 

39. 12(3 5y) 12(3) 12(5y) 36 60y 60y 36 

40. 8(4 7w) 8(4) 8(7w) 32 56w 56w 32 



Chapter 3 




41. c 12 23 

c 12 12 23 12 

c 11 

Check: c 12 23 

11 12 23 

23 23 ✓ 

42. b 14 91 

b 14 14 91 14 

b 77 

Check: b 14 91 

77 14 91 

91 91 ✓ 

43. x 17 45 

x 17 17 45 17 

x 28 

Check: x 17 45 

28 17 45 

45 45 ✓ 

44. d 22 43 

d 22 22 43 22 

d 21 

Check: d 22 43 

21 22 43 

43 43 ✓ 

3.1 Standardized Test Practice (p. 124) 

45. B; 15y 63 57 

15y 63 63 57 63 

15y 120 

1 5y 

1 20 

 

15 

15 

y 8 

46. Let x the amount of each monthly payment. Write a 

verbal model. 

Cost of 

Number of Cost of 

Down 

video monthly p monthly 

payment 

game 

payments payment 

150 25 5x 

Subtract 25 from each side. 

150 25 25 5x 25 

125 5x 

Divide each side by 5. 

12 5 

5 x 

 

5 5 

25 x 

Each monthly payment is $25. 

Lesson 3.2 

3.2 Checkpoint (p. 126) 

1. 3n 40 2n 15 

5n 40 15 

5n 40 40 15 40 

5n 55 

5 n 

5 5 

 

5 5 

n 11 

Check: 3n 40 2n 15 

3(11) 40 2(11) 15 

15 15 ✓ 

2. 2(s 1) 6 

2s 2 6 

2s 2 2 6 2 

2s 8 

2 s 

8 2 2 

s 4 

Check: 2(s 1) 6 

2(4 1) 6 

6 6 ✓ 

3. 13 2y 3(y 4) 

13 2y 3y 12 

13 y 12 

13 12 y 12 12 

25 y 

25 y 

 

1 1 

25 y 

Check: 13 2y 3(y 4) 

13 2(25) 3[(25) 4] 

13 13 ✓ 


1. Use the distributive property to rewrite 6(x 1) 12 as 

6x 6 12. 

2. In the equation 3x 5 2x 8 12, the like terms 

are 3x and 2x, and 5 and 8. 

3. 4 x 7 10 

x 11 10 

x 11 11 10 11 

x 1 

Check: 4 x 7 10 

4 (1) 7 10 

10 10 ✓ 






4. 3x 2x 25 

5x 25 

5 x 

2 5 

 

5 5 

x 5 

Check: 3x 2x 25 

3(5) 2(5) 25 

25 25 ✓ 

5. 21 4x 9 x 

21 3x 9 

21 9 3x 9 9 

30 3x 

3 0 

3 x 

 

3 3 

10 x 

Check: 21 4x 9 x 

21 4(10) 9 10 

21 21 ✓ 

6. 3(x 1) 6 

3x 3 6 

3x 3 3 6 3 

3x 3 

3 x 

3 3 3 

x 1 

Check: 3(x 1) 6 

3(1 1) 6 

6 6 ✓ 

7. 16 8(x 1) 

16 8x 8 

16 8 8x 8 8 

24 8x 

2 4 

8 x 

 

8 8 

3 x 

Check: 16 8(x 1) 

16 8(3 1) 

16 16 ✓ 

8. 5 2(x 2) 19 

5 2x 4 19 

1 2x 19 

1 2x 1 19 1 

2x 18 

2 x 

1 8 

 

2 2 

x 9 



Check: 5 2(x 2) 19 

5 2(9 2) 19 

19 19 ✓ 

9. (1) P 2l 2w 

28 2(10) 2(x 2) 

(2) 28 20 2x 4 

28 24 2x 

28 24 24 2x 24 

4 2x 

4 2 2 x 

 

2 

2 x 

(3) Width x 2 2 2 4 

The width is 4 units. 

(4) Check: P 2l 2w 2(10) 2(4) 20 8 28 

The solutions checks. 


10. Sample answer: The distributive property was not 

applied correctly. 

Distribute the 2 to the entire quantity. 

2(5 n) 2 

10 2n 2 

10 2n 10 2 10 

2n 12 

n 6 

11. 13t 7 10t 2 

3t 7 2 

3t 7 7 2 7 

3t 9 

3 t 

9 3 3 

t 3 

Check: 13t 7 10t 2 

13(3) 7 10(3) 2 

2 2 ✓ 

12. 22 4y 14 0 

8 4y 0 

8 4y 8 0 8 

4y 8 

4 y 

8 

 

4 4 

y 2 

Check: 22 4y 14 0 

22 4(2) 14 0 

0 0 ✓ 



Chapter 3 




13. 2d 24 3d 84 

5d 24 84 

5d 24 24 84 24 

5d 60 

5 d 

6 0 

 

5 5 

d 12 

Check: 2d 24 3d 84 

2(12) 24 3(12) 84 

84 84 ✓ 

14. 4(x 5) 16 

4x 20 16 

4x 20 20 16 20 

4x 4 

4 x 

4 

 

4 4 

x 1 

Check: 4(x 5) 16 

4[(1) 5] 16 

16 16 ✓ 

15. 3(7 2y) 9 

21 6y 9 

21 6y 21 9 21 

6y 12 

6 

 

y 

6 

12 

 

6 

y 2 

Check: 3(7 2y) 9 

3[7 2(2)] 9 

9 9 ✓ 

16. 2(z 11) 6 

2z 22 6 

2z 22 22 6 22 

2z 28 

2z 

28 

 

2 

2 

z 14 

Check: 2(z 11) 6 

2[(14) 11] 6 

6 6 ✓ 

17. 5(3n 5) 20 

15n 25 20 

15n 25 25 20 25 

15n 45 

15n 

45 

 

15 

15 

n 3 





Check: 5(3n 5) 20 

5[3(3) 5] 20 

20 20 ✓ 

18. 30 6( f 5) 

30 6f 30 

30 30 6f 30 30 

0 6f 

0 6 6 f 

 

6 

0 f 

Check: 30 6( f 5) 

30 6(0 5) 

30 30 ✓ 

19. 12 3(m 17) 

12 3m 51 

12 51 3m 51 51 

63 3m 

6 3 

3 m 

3 3 

21 m 

Check: 12 3(m 17) 

12 3(21 17) 

12 12 ✓ 

20. Let x the amount spent on each rod. 

Total Cost of Number Amount spent 

p 

amount licenses of rods on each rod 

200 20 5x 

200 20 20 5x 20 

180 5x 

18 0 

5 x 

 

5 5 

36 x 

They can spend a maximum of $36 on each rod. 

21. Let x the number of people in the group. 

Total Cost for Room cost Number 

p 

cost room per person of people 

Snack cost Number 

 

p 

per person of people 

65 30 5x 2x 

65 30 7x 

65 30 30 7x 30 

35 7x 

3 5 

7 x 

 

7 7 

5 x 

5 people can be in the group. 




22. 5(2w 1) 25 

10w 5 25 

10w 5 5 25 5 

10w 30 

10w 

30 

 

10 

10 

w 3 

Check: 5(2w 1) 25 

5[2(3) 1] 25 

25 25 ✓ 

23. 4(5 p) 8 

20 4p 8 

20 4p 20 8 20 

4p 12 

4 

 

p 

4 

12 

 

4 

p 3 

Check: 4(5 p) 8 

4(5 3) 8 

8 8 ✓ 

24. 40 (2x 5) 61 

40 2x 5 61 

45 2x 61 

45 2x 45 61 45 

2x 16 

2 

 

x 

2 

16 

 

2 

x 8 

Check: 40 (2x 5) 61 

40 [2(8) 5] 61 

61 61 ✓ 

25. 2 4(3k 8) 11k 

2 12k 32 11k 

2 k 32 

2 32 k 32 32 

34 k 

Check: 2 4(3k 8) 11k 

2 4[3(34) 8] 11(34) 

2 2 ✓ 

26. 42 18t 4(t 5) 

42 18t 4t 20 

42 22t 20 

42 20 22t 20 20 

22 22t 

2 2 

2 2t 

 

22 

22 

1 t 





Check: 42 18t 4(t 5) 

42 18(1) 4(1 5) 

42 42 ✓ 

27. 3(2z 8) 10z 16 

6z 24 10z 16 

4z 24 16 

4z 24 24 16 24 

4z 8 

Chapter 3 

4 z 

8 

 

4 4 

z 2 

Check: 3(2z 8) 10z 16 

3[2(2) 8] 10(2) 16 

16 16 ✓ 

28. 5g (8 g) 12 

5g 8 g 12 

4g 8 12 

4g 8 8 12 8 

4g 20 

4g 

20 

 

4 

4 

g 5 

Check: 5g (8 g) 12 

5(5) [8 (5)] 12 

12 12 ✓ 

29. 5 0.25(4 20r) 8r 

5 1 5r 8r 

5 1 3r 

5 1 1 3r 1 

6 3r 

6 

3r 

 

3 

3 

2 r 

Check: 5 0.25(4 20r) 8r 

5 0.25[4 20(2)] 8(2) 

5 5 ✓ 

30. 2m 0.5(m 4) 9 

2m 0.5m 2 9 

2.5m 2 9 

2.5m 2 2 9 2 

2.5m 11 

2 .5m 

11 

 

2.5 

2 . 5 

m 4.4 






Check: 2m 0.5(m 4) 9 

2(4.4) 0.5(4.4 4) 9 

9 9 ✓ 

31. 12 2h 0.2(20 6h) 

12 2h 4 1.2h 

12 3.2h 4 

12 4 3.2h 4 4 

16 3.2h 

16 

3.2h 

 

3. 

2 3.2 

5 h 

Check: 12 2h 0.2(20 6h) 

12 2(5) 0.2[20 6(5)] 

12 12 ✓ 

32. If w the width, then l w 1. 

P 2l 2w 

22 2(w 1) 2w 

22 2w 2 2w 

22 4w 2 

22 2 4w 2 2 

20 4w 

2 0 

4 w 

4 4 

5 w 

The width is 5 inches, and the length is 6 inches. 

33. P 2l 2w 

40 2(x 2) 2(7) 

40 2x 4 14 

40 2x 18 

40 18 2x 18 18 

22 2x 

2 2 

2 x 

 

2 2 

11 x 

34. P a b c 

22 5 x (x 1) 

22 2x 6 

22 6 2x 6 6 

16 2x 

1 6 

2 x 

 

2 2 

8 x 

35. P 4s 

104 4(x 11) 

104 4x 44 

104 44 4x 44 44 

60 4x 

6 0 

4 x 

 

4 4 

15 x 

36. P 2l 2w 

32 2(2x 10) 2(x) 

32 4x 20 2x 

32 6x 20 

32 20 6x 20 20 

12 6x 

1 2 

6 x 

 

6 6 

2 x 

37. a. Let m the total number of minutes you used 

last month. 

Total 

phone 

bill 

29.50 19.50 0.25(m 200) 

b. 29.50 19.50 0.25m 50 

29.50 0.25m 30.5 

29.50 30.5 0.25m 30.5 30.5 

60 0.25m 

60 

0 .25m 

 

0 .25 

0.25 

240 m 

c. 240 200 40, you used 40 additional minutes 

last month. 

38. a. 3(x 7) 42 

3(x 7) 

4 2 

 

3 3 

x 7 14 

x 7 7 14 7 

x 7 

3(x 7) 42 

3x 21 42 

3x 21 21 42 21 

3x 21 

Charge 

Monthly for each 

 

fee 

 

additional 

p 

minute 

3 x 

2 1 

 

3 3 

Number 

of 

minutes 

over 200 

x 7 








b. 4(6x 8) 14 

4(6x 8) 

1 4 

 

4 4 

6x 8 3.5 

6x 8 8 3.5 8 

6x 11.5 

6 x 

11 .5 

 

6 6 

x 1 1 1 

 

12 

4(6x 8) 14 

24x 32 14 

24x 32 32 14 32 

24x 46 

2 4x 

24 

4 6 

2 

4 

x 1 1 1 

 

12 

Sample answer: Both methods produce the same 

answer. Divide first when the number outside the 

parentheses is a factor of the constant on the other side 

of the equation. This method will require fewer steps. 

Distribute first when the number outside the parentheses 

is not a factor of the constant on the other side of the 

equation. This method will be easier because you are 

working with integers. 

39. Total area Area of triangle Area of rectangle 

A 1 bh lw 

2 

1258 1 (3x 1)(24) (3x 1)(25) 

2 

1258 12(3x 1) (3x 1)(25) 

1258 36x 12 75x 25 

1258 111x 37 

1258 37 111x 37 37 

1221 111x 

1 22 

11 

1 

1 

1 11x 

1 

11 

11 x 


40–47. 

Q 

108 6 

4 

R 

J y 

8 

6 

4 

N 

2 

P 

2 4 6 8 x 

M 

K 

4 

L 

6 

8 

40. Begin at the origin. Move 3 units to the left, then 8 units 

up. Point J is located in Quadrant II. 

41. Begin at the origin. Move 8 units to the right, then 3 units 

down. Point K is located in Quadrant IV. 

42. Begin at the origin. Move 4 units to the right, then 4 units 

down. Point L is located in Quadrant IV. 

43. Begin at the origin. Move 1 unit to the left, then 1 unit 

down. Point M is located in Quadrant III. 

44. Begin at the origin. Move 2 units up. Point N is located 

on the y-axis. 

45. Begin at the origin. Move 5 units to the right, then 1 unit 

up. Point P is located in Quadrant I. 

46. Begin at the origin. Move 9 units to the left. Point Q is 

located on the x-axis. 

47. Begin at the origin. Move 5 units to the left, then 8 units 

down. Point R is located in Quadrant III. 

48. a 2 (3 a) a 2 3 a 

a a 2 3 

0 5 

5 

49. 3b 8 2(b 4) 3b 8 2b 8 

3b 2b 8 8 

5b 0 

5b 

50. 2x 5 7(x 1) 2x 5 7x 7 

2x 7x 5 7 

9x (2) 

9x 2 

51. 2y 4 3(y 1) 2y 4 3y 3 

2y 3y 4 3 

5y 1 

52. (2x 3) 4(x 2) 2x 3 4x 8 

2x 4x 3 8 

2x 5 

53. 3(2x 7) 8(4 x) 6x 21 32 8x 

6x 8x 21 32 

2x 11 

54. Let x the number of people. 

Total Cost to Food cost Number 

p 

cost rent space per person of people 

600 150 18x 

600 150 150 18x 150 

450 18x 

4 50 

1 8x 

 

18 

18 

25 x 

So, 25 people can come to the party. 



Chapter 3 





55. D; 3(2x 1) 21 

6x 3 21 

6x 3 3 21 3 

6x 24 

6 

 

x 

6 

24 

 

6 

x 4 

56. Let w the width of the rectangle. 

So, l 2w 5. 

P 2l 2w 

38 2(2w 5) 2w 

38 4w 10 2w 

38 6w 10 

38 10 6w 10 10 

48 6w 

4 8 

6 w 

6 6 

8 w 

So, the width of the rectangle is 8 feet, and the length is 

2(8) 5 16 5 11 feet. 

Brain Game (p. 129) 

A. 10x 7 17 

10x 7 7 17 7 

10x 10 

1 0x 

10 

1 0 

1 

0 

x 1 

2(7x 6) 40 

14x 12 40 

14x 12 12 40 12 

14x 28 

1 4x 

14 

2 8 

1 

4 

x 2 

(x 11) 10 

x 11 10 

x 11 11 10 11 

x 1 

x 

 

1 

1 1 

x 1 


Brain Game ––CONTINUED–– 

B. 8x 15 47 

8x 15 15 47 15 

8x 32 

8 x 

32 

8 8 

x 4 

6(2x 1) 90 

12x 6 90 

12x 6 6 90 6 

12x 96 

1 2x 

9 6 

 

12 

12 

x 8 

7x 4x 24 

3x 24 

3 

 

x 

3 

24 

 

3 

x 8 

C. 5x 4x 6 

x 6 

x 

 

6 

1 1 

x 6 

7x (12) 61 

7x (12) (12) 61 (12) 

7x 49 

7 x 

4 9 

 

7 7 

x 7 

7(x 2) 63 

7x 14 63 

7x 14 14 63 14 

7x 49 

7 x 

4 9 

 

7 7 

x 7 

D. 2(6x 7) 50 

12x 14 50 

12x 14 14 50 14 

12x 36 

1 2x 

3 6 

 

12 

12 

x 3 







Brain Game ––CONTINUED–– 

5x 3x 56 

8x 56 

8 

 

x 

8 

56 

 

8 

x 7 

11x 9 42 

11x 9 9 42 9 

11x 33 

11 

1 

x 

1 

3 

 

3 

11 

x 3 

1 8 7 3 

B C D 

A 

Lesson 3.3 


1. 9 2x 1 3x 

 

3. 5x 2 3x 14 

 

 

 

 

So, the solution is 6. 

4. Yes. Sample answer: Removing x-tiles has no effect on 

the 1-tiles and vice versa. 

5. Sample answer: Solving an equation with variables on 

both sides of the equal sign adds one extra step, because 

you must first get all variable terms on one side. 


2. 3x 4 8 x 

 

 

 

 


 

 

 


1. 5n 2 3n 6 

5n 2 3n 3n 6 3n 

2n 2 6 

2n 2 2 6 2 

2n 8 

2 n 

8 2 2 

n 4 

Check: 5n 2 3n 6 

5(4) 2 3(4) 6 

18 18 ✓ 

2. 8y 4 11y 17 

8y 4 8y 11y 17 8y 

4 3y 17 

4 17 3y 17 17 

21 3y 

2 1 

3 y 

 

3 3 

7 y 

Check: 8y 4 11y 17 

8(7) 4 11(7) 17 

60 60 ✓ 



Chapter 3 




3. m 1 9m 15 

m 1 m 9m 15 m 

1 8m 15 

1 15 8m 15 15 

16 8m 

16 8m 

8 8 

2 m 

Check: m 1 9m 15 

2 1 9(2) 15 

3 3 ✓ 


1. Sample answer: To solve 8x 5 2x 7, subtract 2x 

from each side. 

8x 5 2x 2x 7 2x 

6x 5 7 

Subtract 5 from each side. 

6x 5 5 7 5 

6x 12 

Divide each side by 6. 

6 x 

12 

6 6 

x 2 

2. Sample answer: Continue solving the equation. 

5z 2 5z 

5z 2 5z 5z 5z 

2 0 

This is a false statement, so the equation has no solution. 

3. 13m 22 9m 6 

13m 22 9m 9m 6 9m 

4m 22 6 

4m 22 22 6 22 

4m 16 

4 m 16 

 

4 4 

m 4 

Check: 13m 22 9m 6 

13(4) 22 9(4) 6 

30 30 ✓ 

4. 19c 26 41 14c 

19c 26 14c 41 14c 14c 

5c 26 41 

5c 26 26 41 26 

5c 15 

5 c 

1 5 

 

5 5 

c 3 

Check: 19c 26 41 14c 

19(3) 26 41 14(3) 

83 83 ✓ 

5. 15 4x 42 7x 

15 4x 7x 42 7x 7x 

15 3x 42 

15 3x 15 42 15 

3x 27 

3 x 

2 7 

 

3 3 

x 9 

Check: 15 4x 42 7x 

15 4(9) 42 7(9) 

21 21 ✓ 

6. 14 5y 50 4y 

14 5y 4y 50 4y 4y 

14 9y 50 

14 9y 14 50 14 

9y 36 

9 y 

3 6 

 

9 9 

y 4 

Check: 14 5y 50 4y 

14 5(4) 50 4(4) 

34 34 ✓ 

7. 18w 2 10w 14 

18w 2 10w 10w 14 10w 

8w 2 14 

8w 2 2 14 2 

8w 16 

8 w 16 

 

8 8 

w 2 

Check: 18w 2 10w 14 

18(2) 2 10(2) 14 

34 34 ✓ 






8. 5a 6 6a 38 

5a 6 5a 6a 38 5a 

6 11a 38 

6 38 11a 38 38 

44 11a 

4 4 

1 1a 

 

11 11 

4 a 

Check: 5a 6 6a 38 

5(4) 6 6(4) 38 

14 14 ✓ 

9. Sample answer: Subtracting 4x from x yields 3x, not 3x. 

4x 7 x 2 

4x 7 4x x 2 4x 

7 3x 2 

7 2 3x 2 2 

9 3x 

9 

3x 

3 3 

3 x 

10. Let x the cost of a DVD. 

Your 

cost of 

clothes 

Number 

Number Cost Friend’s 

of 

of your p of a cost of 

friend’s 

p 

DVDs DVD clothes 

DVDs 

Each DVD costs $12. 

60 3x 0 8x 

60 3x 3x 0 8x 3x 

60 5x 

6 0 

5 x 

 

5 5 

12 x 


11. 25u 74 23u 92 

25u 74 23u 23u 92 23u 

2u 74 92 

2u 74 74 92 74 

2u 18 

2 u 

1 8 

 

2 2 

Cost 

of a 

DVD 

12. 5k 19 5 13k 

5k 19 13k 5 13k 13k 

8k 19 5 

8k 19 19 5 19 

8k 24 

8 k 

2 4 

 

8 8 

k 3 

Check: 5k 19 5 13k 

5(3) 19 5 13(3) 

34 34 ✓ 

13. 11y 32 104 5y 

11y 32 11y 104 5y 11y 

32 104 6y 

32 104 104 6y 104 

72 6y 

72 6y 

 

6 6 

12 y 

Check: 11y 32 104 5y 

11(12) 32 104 5(12) 

164 164 ✓ 

14. 15n 16 86 29n 

15n 16 29n 86 29n 29n 

14n 16 86 

14n 16 16 86 16 

14n 70 

1 4n 

7 0 

 

14 

14 

n 5 

Check: 15n 16 86 29n 

15(5) 16 86 29(5) 

59 59 ✓ 

15. 25t 5(5t 1) 

25t 25t 5 

25t 25t 25t 5 25t 

0 5 

No solution 

u 9 

Check: 25u 74 23u 92 

25(9) 74 23(9) 92 

299 299 ✓ 



Chapter 3 




16. 13 3p 5(3 2p) 

13 3p 15 10p 

13 3p 10p 15 10p 10p 

13 7p 15 

13 7p 13 15 13 

7p 28 

7 p 

28 

7 7 

p 4 

Check: 13 3p 5(3 2p) 

13 3(4) 5[3 2(4)] 

25 25 ✓ 

17. 24s 53 39 s 

24s 53 24s 39 s 24s 

53 39 23s 

53 39 39 23s 39 

92 23s 

92 

23 

2 3s 

2 

3 

4 s 

Check: 24s 53 39 s 

24(4) 53 39 (4) 

43 43 ✓ 

18. 14a 93 49 57a 

14a 93 57a 49 57a 57a 

71a 93 49 

71a 93 93 49 93 

71a 142 

7 1a 

1 42 

 

71 

71 

a 2 

Check: 14a 93 49 57a 

14(2) 93 49 57(2) 

65 65 ✓ 

19. 7(2p 1) 14p 7 

14p 7 14p 7 

Every number is a solution. 

20. 8v 2(4v 2) 

8v 8v 4 

8v 8v 8v 4 8v 

0 4 

No solution 

21. 3x 6 3(2 x) 

3x 6 6 3x 


22. 2(4h 13) 37 13h 

8h 26 37 13h 

8h 26 8h 37 13h 8h 

26 37 21h 

26 37 37 21h 37 

63 21h 

6 

2 

3 

1 

2 1h 

2 

1 

3 h 

Check: 2(4h 13) 37 13h 

2[4(3) 13] 37 13(3) 

2 2 ✓ 

23. 9 2x 3x 2 

9 2x 2x 3x 2 2x 

9 x 2 

9 2 x 2 2 

11 x 

24. 11x 3 9 5x 

11x 3 5x 9 5x 5x 

6x 3 9 

6x 3 3 9 3 

6x 12 

6 x 

1 2 

 

6 6 

x 2 

25. 4 7x 12 3x 

4 7x 7x 12 3x 7x 

4 12 4x 

4 12 12 4x 12 

8 4x 

8 

4 x 

 

4 4 

2 x 

26. 9x 12 8 4x 

9x 12 9x 8 4x 9x 

12 8 5x 

12 8 8 5x 8 

20 5x 

20 5x 

 

5 5 

4 x 






27. Let x the number of times you pay for the tolls. 

Toll cost Times Cost to 

Times 

Toll cost 

without p you pay replace p you pay 

with tag 

tag tolls tag 

tolls 

3x 24 1.5x 

3x 1.5x 24 1.5x 1.5x 

1.5x 24 

1 . 5x 

24 

 

1. 

5 1 . 5 

x 16 

You will have to use the tag 16 times for the cost to be 

the same. 

28. 4x 36 5x 

4x 5x 36 5x 5x 

9x 36 

9 x 

3 6 

 

9 9 

x 4 

4x 4(4) 16 

P 4s 4(16) 64 

The perimeter of the square is 64 units. 

29. 12x 7x 30 

12x 7x 7x 30 7x 

5x 30 

5 x 

3 0 

 

5 5 

x 6 

12x 12(6) 72 

P 4s 4(72) 288 


30. 9x 5x 32 

9x 5x 5x 32 5x 

4x 32 

4 x 

3 2 

 

4 4 

x 8 

9x 9(8) 72 

P 4s 4(72) 288 


31. a. 700 60x 

b. 400 60x 

c. 700 60x 400 60x 

700 60x 60x 400 60x 60x 

700 400 120x 

700 400 400 120x 400 

300 120x 

3 00 

1 20x 

 

120 

120 

2.5 x 

In 2.5 hours, you will be exactly halfway between 

Houston and home. 

d. 700 45x 400 45x 

700 45x 45x 400 45x 45x 

700 400 90x 

700 400 400 90x 400 

300 90x 

3 00 

9 0x 

 

90 

90 

3 1 3 x 

You will drive 3 1 3 hours before you are exactly 

halfway between Houston and home. 

32. Let x the batches of pasta you make. 

Cost of 

purchased 

pasta 

Cost of Cost to 

Batches 

p pasta make p 

of pasta 

machine pasta 

0.99x 33 0.33x 

0.99x 0.33x 33 0.33x 0.33x 

0.66x 33 

0 . 66x 

33 

 

0. 

66 

0 .66 

x 50 

Batches 

of pasta 

You will need to make 50 batches of pasta for the costs to 

be equal. 



Chapter 3 




33. Sample answer: You and a friend decide to start a dogwalking 

service. In one day you walk 11 dogs and earn 

$5 in tips. Your friend walks 8 dogs and earns $23 in tips. 

If you both earn the same amount of money, what was the 

cost to walk a dog? 

Let x the cost to walk a dog. 

Dogs 

you 

walked 

Dogs 

Cost to 

Cost to 

Your your 

p walk a p walk a 

tips friend 

 

dog 

dog 

walked 

11x 5 8x 23 

11x 5 8x 8x 23 8x 

3x 5 23 

3x 5 5 23 5 

3x 18 

3 x 

1 8 

 

3 3 

x 6 

$6 was charged to walk a dog. 

34. 3x 7 8 6(x 2) 

3x 7 8 6x 12 

3x 7 6x 20 

3x 7 3x 6x 20 3x 

7 3x 20 

7 20 3x 20 20 

27 3x 

27 3x 

 

3 3 

9 x 

Check: 3x 7 8 6(x 2) 

3(9) 7 8 6(9 2) 

34 34 ✓ 

35. 13y 19 6(9 y) 14 

13y 19 54 6y 14 

13y 19 6y 68 

13y 19 6y 6y 68 6y 

7y 19 68 

7y 19 19 68 19 

7y 49 

7 y 

4 9 

 

7 7 

y 7 

Check: 13y 19 6(9 y) 14 

13(7) 19 6(9 7) 14 

110 110 ✓ 

Your 

friend’s 

tips 

36. 8(z 4) 5(13 z) 

8z 32 65 5z 

8z 32 5z 65 5z 5z 

3z 32 65 

3z 32 32 65 32 

3z 33 

3 z 

3 3 

 

3 3 

z 11 

Check: 8(z 4) 5(13 z) 

8(11 4) 5(13 11) 

120 120 ✓ 

37. 8a 2(a 5) 2(a 1) 

8a 2a 10 2a 2 

6a 10 2a 2 

6a 10 2a 2a 2 2a 

4a 10 2 

4a 10 10 2 10 

4a 8 

4 a 

8 4 4 

a 2 

Check: 8a 2(a 5) 2(a 1) 

8(2) 2(2 5) 2(2 1) 

2 2 ✓ 

38. a. The triangle has the greater side length. Sample answer: 

The triangle has the greater side length because the 

sum of 3 of its sides is equal to the sum of 4 of the 

square’s sides. 

b. 3x 2x 3 

3x 2x 2x 3 2x 

x 3 

3x 3(3) 9 

The side length of the square is 9 units. 

5x 3 5(3) 3 15 3 12 

The side length of the triangle is 12 units. 

c. P 4s 4(9) 36 


P 3s 3(12) 36 

The perimeter of the triangle is 36 units. 






39. 0.75m 14 1.87m 10.3936 

0.75m 14 0.75m 1.87m 10.3936 0.75m 

14 1.12m 10.3936 

14 10.3936 1.12m 10.3936 10.3936 

24.3936 1.12m 

24 .39 

1.1 

36 

2 

1 .12m 

1 

.12 

21.78 m 

Check: 0.75m 14 1.87m 10.3936 

0.75(21.78) 14 1.87(21.78) 10.3936 

30.335 30.335 ✓ 

40. 19.5 0.5t 10.6206 0.4t 

19.5 0.5t 0.4t 10.6206 0.4t 0.4t 

19.5 0.9t 10.6206 

19.5 0.9t 19.5 10.6206 19.5 

0.9t 8.8794 

0 . 9 

0. 

t 

9 

8 .8794 

0 

.9 

t 9.866 

Check: 19.5 0.5t 10.6206 0.4t 

19.5 0.5(9.866) 10.6206 0.4(9.866) 

14.567 14.567 ✓ 

41. 9.39 3.4d 1.1d 11.08 

9.39 3.4d 3.4d 1.1d 11.08 3.4d 

9.39 2.3d 11.08 

9.39 11.08 2.3d 11.08 11.08 

20.47 2.3d 

2 0.47 

2.3 

2 .3d 

2 

.3 

8.9 d 

Check: 9.39 3.4d 1.1d 11.08 

9.39 3.4(8.9) 1.1(8.9) 11.08 

20.87 20.87 ✓ 

42. 130.5 9b 55.104 3.2b 

130.5 9b 9b 55.104 3.2b 9b 

130.5 55.104 12.2b 

130.5 55.104 55.104 12.2b 55.104 

75.396 12.2b 

7 5. 

12 

396 

. 2 

1 2.2b 

1 

2.2 

6.18 b 

Check: 130.5 9b 55.104 3.2b 

130.5 9(6.18) 55.104 3.2(6.18) 

74.88 74.88 ✓ 

43. a. ax 6 2(x 3) 

ax 6 2x 6 

When a 2, the equation has all numbers as 

a solution. 

b. The equation has just one solution when a equals any 

number except 2. 


44. c 20 14 

c 20 20 14 20 

c 34 

Check: c 20 14 

34 20 14 

14 14 ✓ 

45. d 9 12 

d 9 9 12 9 

d 21 

Check: d 9 12 

21 9 12 

12 12 ✓ 

46. x 3 17 

x 3 3 17 3 

x 20 

Check: x 3 17 

20 3 17 

17 17 ✓ 

47. y 21 15 

y 21 21 15 21 

y 6 

Check: y 21 15 

6 21 15 

15 15 ✓ 

48. Let x the number of months. 

Total One-time Cost per Number 

p 

cost fee month of months 

345 75 45x 

345 75 75 45x 75 

270 45x 

2 70 

4 5x 

 

45 

45 

6 x 

Your friend has been a member of the gym for 6 months. 



Chapter 3 




49. P 4s 

32 4(x 3) 

32 4x 12 

32 12 4x 12 12 

20 4x 

2 0 

4 x 

 

4 4 

5 x 


50. D; 2(3x 4) 6x 5 

6x 8 6x 5 

6x 8 6x 6x 5 6x 

8 6 

No solution 

51. F; 2y 7 11 5y 

2(6) 7 11 5(6) 

12 7 11 30 

19 19 

3.3 Technology Activity (p. 136) 

1. x 2 2x 6 

The values are the same when x 4. 

2. 3x 1 x 7 


3. 12 x x 4 


4. 7x 16 x 


5. 5x 2 8x 1 


6. 4x 6 2x 4 


7. 3x 6 13x 2 

3x 6 3x 13x 2 3x 

6 10x 2 

6 2 10x 2 2 

4 10x 

4 

1 0x 

1 0 10 

0.4 x 

Sample answer: To solve the equation using a calculator, 

change TTbl from 1 to 0.1. 

Mid-Chapter Quiz (p. 137) 

1. 2x 5 27 

2x 5 5 27 5 

2x 22 

2 x 

2 2 

 

2 2 

x 11 

2. 7(4 x) 14 

28 7x 14 

28 7x 28 14 28 

7x 42 

7 x 

42 

7 7 

x 6 

3. 4x 3 2x 9 

4x 3 2x 2x 9 2x 

2x 3 9 

2x 3 3 9 3 

2x 12 

2 x 

12 

2 2 

x 6 

4. 11k 9 42 

11k 9 9 42 9 

11k 33 

1 1k 

11 

3 3 

1 

1 

k 3 

Check: 11k 9 42 

11(3) 9 42 

42 42 ✓ 

5. a 11 5 

3 

a 11 11 5 11 

3 

Check: 

a 3 16 

3 a 3 3(16) 

a 48 

a 11 5 

3 

48 11 5 

3 

5 5 ✓ 






6. w 18 7 

2 

w 18 18 7 18 

2 

w 2 11 

2 w 2 2(11) 

w 22 

Check: w 18 7 

2 

2 2 

18 

2 

7 

7 7 ✓ 

7. 2 5t 3 34 

5t 1 34 

5t 1 1 34 1 

5t 35 

5 t 

3 5 

 

5 5 

t 7 

Check: 2 5t 3 34 

2 5(7) 3 34 

34 34 ✓ 

8. 3y 15 y 39 

4y 15 39 

4y 15 15 39 15 

4y 24 

4y 

24 

 

4 

4 

y 6 

Check: 3y 15 y 39 

3(6) 15 (6) 39 

39 39 ✓ 

9. 5(n 2) 10 

5n 10 10 

5n 10 10 10 10 

5n 0 

5 n 

0 5 5 

n 0 

Check: 5(n 2) 10 

5(0 2) 10 

10 10 ✓ 

10. 2 5(h 3) 28 

2 5h 15 28 

5h 13 28 

5h 13 13 28 13 

5h 15 

5 

 

h 

5 

15 

 

5 

h 3 

Check: 2 5(h 3) 28 

2 5(3 3) 28 

28 28 ✓ 

11. 5s 7s 1 2s 

5s 5s 1 

5s 5s 5s 1 5s 

0 1 

No solution 

12. 4d 5 d 

4d 5 4d d 4d 

5 5d 

5 

5d 

 

5 

5 

1 d 

Check: 4d 5 d 

4(1) 5 1 

1 1 ✓ 

13. 17 5m 50 6m 

17 5m 5m 50 6m 5m 

17 50 11m 

17 50 50 11m 50 

33 11m 

33 

11 

1 1m 

1 

1 

3 m 

Check: 17 5m 50 6m 

17 5(3) 50 6(3) 

32 32 ✓ 

14. 3f 12 3( f 12) 

3f 12 3f 36 

3f 12 3f 3f 36 3f 

12 36 

No solution 

15. 8(4p 1) 32p 8 

32p 8 32p 8 




Chapter 3 




16. Let x the hourly wage. 

Hours 

Hours on Hourly Tips on 

Hourly 

p on p 

Wednesday wage Wednesday 

wage 

Friday 

Tips 

on 

Friday 

5x 25 3x 30.76 

5x 25 3x 3x 30.76 3x 

2x 25 30.76 

2x 25 25 30.76 25 

2x 5.76 

2 x 

5. 76 

2 2 

x 2.88 

Your friend’s hourly wage is $2.88. 

17. 4x 3 2x 5 

4x 3 2x 2x 5 2x 

2x 3 5 

2x 3 3 5 3 

2x 8 

2 x 

8 2 2 

Lesson 3.4 


1. n 7 > 3 

n 7 7 > 3 7 

n > 4 

6 5 4 3 2 1 

Check: n 7 > 3 

1 7 ? > 3 

6 > 3 ✓ 

2. 10 ≥ y 4 

10 4 ≥ y 4 4 

6 ≥ y, ory ≤ 6 

0 2 4 6 8 10 

Check: 10 ≥ y 4 

10 ? ≥ 3 4 

10 ≥ 7 ✓ 

3. 6 ≤ x 9 

6 9 ≤ x 9 9 

3 ≤ x, orx ≥ 3 

0 

12 

x 4 

2x 5 2(4) 5 8 5 13 

P 3s 3(13) 39 

The perimeter of the triangle is 39 units. 


Let x the number of small boxes in each large box. 

Large 

boxes 

Small 

Small 

boxes Unpacked boxes 

Large 

p in each small p in each 

boxes 

 

large boxes 

large 

box 

box 

3x 24 5x 10 

3x 24 3x 5x 10 3x 

24 2x 10 

24 10 2x 10 10 

14 2x 

1 4 

2 x 

 

2 2 

7 x 

Each large box holds 7 small boxes. 

3x 24 3(7) 24 21 24 45 

Each person will pack 45 small boxes. 

45 7 ≈ 6.4 

Each person will need 7 large boxes to pack all of the 

small boxes. 

Unpacked 

small 

boxes 

1 

0 1 2 3 4 

Check: 6 ≤ x 9 

6 ? ≤ 5 9 

6 ≤ 4 ✓ 

4. z 5 < 1 

z 5 5 < 1 5 

z < 6 

0 2 4 6 8 10 

Check: z 5 < 1 

4 5 ? < 1 

1 < 1 ✓ 


1. Equivalent inequalities are inequalities that have the 

same solution. 

2. Sample answer: The graph of x > 5 has an open circle at 

5 because 5 is not part of the solution. The graph of x ≥ 5 

has a closed circle at 5 because 5 is part of the solution. 

3. 5 < n 

5 < 8, so 8 is a solution. 

4. 5 < n 

5 > 8, so 8 is not a solution. 

12 

5. 5 < n 


5 






6. 5 < n 


7. x 2 > 3 

x 2 2 > 3 2 

x > 5 

6 

Check: x 2 > 3 

2 2 ? > 3 

0 > 3 ✓ 

8. 1 ≥ x 9 

1 9 ≥ x 9 9 

10 ≥ x, orx ≤ 10 

Check: 1 ≥ x 9 

1 ? ≥ 8 9 

1 ≥ 1 ✓ 

9. x 4 < 3 

x 4 4 < 3 4 

x < 1 

3 

5 

0 2 4 6 8 10 

2 

Check: x 4 < 3 

2 4 ? < 3 

2 < 3 ✓ 

10. x 3 > 7 

x 3 4 > 7 3 

x > 4 

0 1 2 3 4 5 

Check: x 3 > 7 

6 3 ? > 7 

9 > 7 ✓ 

11. (1) Let x the number of hours the pilot must log. 

Hours pilot 

must log 

 

x 250 ≥ 1000 

(2) x 250 250 ≥ 1000 250 

x ≥ 750 

0 

4 

1 

3 

0 1 2 3 

500 

2 

1 

Hours 

logged 

1000 

0 

12 

6 

≥ 

Minimum 

hours needed 

1500 

Check: x 250 ≥ 1000 

753 250 ? ≥ 1000 

1003 ≥ 1000 ✓ 

(3) The pilot must log at least 750 more hours to become 

a pilot astronaut. 


12. Let t the weight a forklift can raise, in pounds. 

t ≤ 2500 

13. Let s the speed limit, in miles per hour. 

s ≤ 55 

14. Let w the weight a truck can tow, in pounds. 

w ≤ 7700 

15. Let h your height, in inches. 

h ≥ 48 

16. Let s the savings on DVD players, in dollars. 

s ≤ 50 

17. x > 1 18. x ≥ 5 19. x ≤ 6 20. x < 20 

21. x 4 < 5 

x 4 4 < 5 4 

x < 1 

3 2 1 

22. m 8 ≥ 12 

m 8 8 ≥ 12 8 

m ≥ 4 

23. 11 < y 5 

11 5 < y 5 5 

16 < y, ory > 16 

24 20 16 12 8 

24. 8 ≥ d 7 

8 7 ≥ d 7 7 

1 ≥ d, ord ≤ 1 

3 

2 

25. 45 > g 16 

45 16 > g 16 16 

29 > g, org < 29 

26. z 15 > 72 

z 15 15 > 72 15 

z > 87 

27. f 1 ≥ 8 

f 1 1 ≥ 8 1 

f ≥ 9 

15 12 

1 

31 29 27 25 

83 84 85 

9 

86 87 88 89 

6 

0 1 2 3 

0 1 2 3 4 5 

0 1 2 3 

3 

4 

6 

0 

0 3 



Chapter 3 




28. h 19 ≤ 15 

h 19 19 ≤ 15 19 

h ≤ 4 

6 5 4 3 2 1 

29. 18.1 ≤ p 7 

18.1 7 ≤ p 7 7 

25.1 ≤ p, orp ≥ 25.1 

0 

37. q 4 16 ≥ 30 

q 20 ≥ 30 

q 20 20 ≥ 30 20 

q ≥ 10 

5 

0 5 10 15 20 25 

38. Let x the temperature in F at which the bacteria 

can survive. 

x ≥ 280 

25 25.2 25.4 25.6 25.8 26 

30. t 7 < 3.4 

t 7 7 < 3.4 7 

t < 10.4 

10 10.2 10.4 10.6 10.8 11 

31. b 2.5 ≤ 2.5 

b 2.5 2.5 ≤ 2.5 2.5 

b ≤ 0 

3 2 1 0 1 2 3 

32. a 10.2 > 5.3 

a 10.2 10.2 > 5.3 10.2 

a > 15.5 

15 15.2 15.4 15.6 15.8 16 

33. Let x the temperature in F at which neon is a gas. 

x ≥ 411 

413 411 409 407 

34. 5 m 8 ≥ 14 

m 13 ≥ 14 

m 13 13 ≥ 14 13 

m ≥ 1 

300 

280 

39. No. Sample answer: It is not possible to check all the 

numbers that are solutions of an inequality because a 

solution consists of an infinite amount of numbers. 

No. Sample answer: Checking one number does not 

guarantee that a solution is correct. 

40. a. Let w the weight in pounds you can add to the 

first bag. 

Weight you 

can add 

 

≤ 

w 14 ≤ 50 

w 14 14 ≤ 50 14 

w ≤ 36 

You can add at most 36 pounds of personal belongings 

to the first bag. 

b. Let w the weight in pounds you can add to the 

second bag. 

Weight you 

can add 

260 

 

Weight 

of bag 

Weight 

of bag 

240 

≤ 

Weight 

limit 

Weight 

limit 

w 21 ≤ 50 

w 21 21 ≤ 50 21 

w ≤ 29 

You can add at most 29 pounds of personal belongings 

to the second bag. 

41. x ≥ 1 and x ≤ 4 

3 

2 

1 

0 1 2 3 

1 

0 1 2 3 4 5 

35. 13 n 26 < 38 

n 13 < 38 

n 13 13 < 38 13 

n < 51 

47 

48 

49 

50 51 52 53 

36. 2.35 p 14.9 > 49.25 

p 17.25 > 49.25 

p 17.25 17.25 > 49.25 17.25 

p > 32 

42. x < 3 and x ≥ 0 

1 

43. Let x the temperature of ski wax in C. 

x ≥ 6 

x ≤ 15 

x ≥ 6 and x ≤ 15 

6 

10 5 

0 1 2 3 4 5 

0 5 10 15 20 

0 

8 

16 

24 32 40 48 






44. Sample answer: Place a closed circle at 8 and shade 

the number line to the left. Also, place a closed circle at 

10 and shade the number line to the right. This graph 

consists of two arrows, one heading left and one heading 

right with a gap in between them, while the graph of 

x ≥ 8 and x ≤ 10 is just the portion of the number line 

between the points 8 and 10 with closed circles at 8 

and 10. 


45. P 4s 

36.6 4s 

36 .6 

4 s 

 

4 4 

9.15 s 

The square has a side length of 9.15 meters. 

46. Let x the number of uniforms. 

Total Cost of Cost of Number of 

p 

amount equipment uniform uniforms 

1275 450 55x 

1275 450 450 55x 450 

825 55x 

8 25 

5 5x 

 

55 

55 

15 x 

The team can buy 15 uniforms. 

47. 5 4x 7x 11 

5 4x 4x 7x 11 4x 

5 3x 11 

5 11 3x 11 11 

6 3x 

6 

3 x 

 

3 3 

2 x 

48. 3x 8 3 2x 

3x 8 2x 3 2x 2x 

x 8 3 

x 8 8 3 8 

x 5 

Lesson 3.5 


1. Sample answer: 

(1) 6 < 10 

(2) 2 p (6) ? < 2 p 10 

12 < 20 ✓ 

Yes, 12 is less than 20. 

(3) 2 p (6) ? < 2 p 10 

12 < 20 ✗ 

No, 12 is not less than 20. 

(4) 6 

? < 1 0 

 

2 2 

3 < 5 ✓ 

Yes, 3 is less than 5. 

(5) 6 

? 10 

< 

2 

2 

3 < 5 ✗ 

No, 3 is not less than 5. 

In steps 3 and 5, you could reverse the inequality symbols 

to make the statements true. 

2. a 2 > b 2 3. a b 

< 

2 

2 

4. a 3b 


1. n 6 > 7 

6 p n 6 > 6 p 7 

39 

n > 42 

40 

41 

t 

2. ≤ 8 4 

t 

4 p ≥ 4 p 8 

4 

t ≥ 32 

42 43 44 45 


49. B 

50. F; b 2 > 2 

4 2 ? > 2 

6 > 2 ✓ 

48 32 16 

3. 2x > 8 

2 x 

> 8 

 

2 2 

x > 4 

5 4 3 2 1 

0 

0 1 



Chapter 3 




4. 7s ≤ 14 

7s 

14 

≥ 

7 

7 

s ≥ 2 

5 4 3 2 1 


1. To solve 7y ≤ 49, use the division property of inequality. 

2. Sample answer: To solve 2x > 14, divide each side 

by 2, which is positive, so the direction of the inequality 

symbol does not change. 

To solve 2x > 14, divide each side by 2, which is 

negative, so you must reverse the direction of the 

inequality symbol. 

v 

3. < 8 2 

v 

2 p > 2(8) 

2 

v > 16 

13 14 

v 

Check: < 8 2 

4. 8b > 32 

8 b 

> 3 2 

 

8 8 

b > 4 

0 1 

18 

? < 8 2 

9 < 8 ✓ 

Check: 8b > 32 

8(6) ? > 32 

48 > 32 ✓ 

5. u 6 ≥ 3 

6 p u 6 ≥ 6 p 3 

u ≥ 18 

15 16 

Check: u 6 ≥ 3 

15 16 17 18 19 

2 3 4 5 6 

17 18 19 20 21 

2 1 

? ≥ 3 

6 

3.5 ≥ 3 ✓ 

0 1 

6. 6s ≤ 54 

6s 

54 

≥ 

6 

6 

s ≥ 9 

12 11 10 9 8 7 6 

Check: 6s ≤ 54 

6(6) ? ≤ 54 

36 ≤ 54 ✓ 

7. 5a < 35 

5 a 

< 35 

5 5 

a < 7 

10 9 

Check: 5a < 35 

5(10) ? < 35 

50 < 35 ✓ 

8. p 7 > 6 

7 p p 7 > 7 p 6 

p > 42 

39 40 

Check: p 7 > 6 

9. 3r ≥ 21 

3 r 

≥ 2 1 

 

3 3 

r ≥ 7 

4 9 

? > 6 

7 

7 > 6 ✓ 

4 5 6 7 8 9 10 

Check: 3r ≥ 21 

3(9) ? ≥ 21 

27 ≥ 21 ✓ 

10. 4 

t ≤ 9 

4 p 4 

t ≤ 4(9) 

Check: 

8 

t ≤ 36 

7 

6 

41 42 43 44 45 

39 38 37 36 35 34 33 

4 

t ≤ 9 

5 

4 

38 

? ≤ 9 

4 



9.5 ≤ 9 ✓ 




11. (1) Let c the number of cups of pasta. 

Calories 

per cup 

p 

≥ 

200c ≥ 500 

(2) 2 00 

20 

c 

0 

≥ 5 0 

2 

0 

00 

c ≥ 2.5 

(3) Sample answer: You should eat at least 2.5 cups 

of pasta at one meal to get the desired number 

of calories. 


12. a 2 < 9 

2 p a 2 < 2(9) 

a < 18 

36 24 12 0 

13. b 7 > 7 

7 p b 7 > 7 p 7 

b > 49 

14. 8 

c ≥ 3 

46 47 48 49 50 51 52 

8 p 8 

c ≥ 8 p 3 

0 

c ≥ 24 

6 

12 

15. 16y > 48 

16y 

48 

< 

16 

16 

y < 3 

Number 

of cups 

18 24 30 36 

Total calories 

for meal 

18. 12x ≥ 60 

1 2x 

≥ 60 

 

12 

12 

x ≥ 5 

6 5 4 3 2 1 0 

19. 4w ≤ 68 

4 w 68 

≤ 

4 4 

w ≤ 17 

13 14 15 16 17 18 19 

20. 9 

t < 12 

9 p 9 

t < 9(12) 

t < 108 

112 110 108 106 

h 

21. ≤ 13 6 

h 

6 p ≥ 6 p 13 

6 

h ≥ 78 

80 78 76 74 

22. 16k ≥ 96 

16k 

96 

≤ 

16 

16 

k ≤ 6 

12 10 8 6 4 2 0 

23. 6q > 84 

6 q 

> 84 

6 6 

q > 14 

6 5 4 3 2 1 

16. 5z < 65 

5 z 

< 6 5 

 

5 5 

z < 13 

0 

16 14 12 10 

24. 7s ≥ 84 

7 

 

s 

7 

≤ 84 

 

7 

s ≤ 12 

9 

10 

11 

17. 

d 

11 ≤ 6 

12 13 14 15 

d 

11 p 11 ≥ 11 p 6 

d ≥ 66 

0 3 6 9 12 15 18 

25. 4m < 60 

4 m 60 < 

4 4 

m < 15 

30 20 10 0 

68 66 64 62 



Chapter 3 




26. 5 

v > 2 

5 p 5 

v > 5(2) 

v > 10 

20 15 10 5 0 5 10 

n 

27. ≥ 5 3 

n 

3 p ≤ 3(5) 

3 

n ≤ 15 

5 0 5 10 15 20 25 

28. Do not reverse the inequality symbol unless you are 

dividing by a negative number. 

9x > 45 

9 x 

> 45 

9 9 

x > 5 

29. Let x the number of times you use the in-line skates. 

Cost of Cost to rent Times you use 

< p 

in-line skates in-line skates in-line skates 

60 < 12x 

6 0 

< 1 2x 

 

12 

12 

5 < x 

You will have to use the in-line skates more than 5 times. 

30. 5x ≥ 45 31. 4 

x ≤ 8 

5 x 

≥ 4 5 

 

5 5 

x ≥ 9 

4 p 4 

x ≤ 4 p 8 

x ≤ 32 

x 

32. < 6 33. 7x > 35 

3 

x 

3 p > 3 p 6 

3 

x > 18 

7 x 

> 35 

7 7 

x > 5 

x 

34. ≤ 5 35. 3x > 18 

2 

x 3 x 

> 18 

2 p ≤ 2 p 5 2 

3 3 

x > 6 

x ≤ 10 

36. a. Let x the number of crates. 

Weight 

of crate 

p 

≤ 

375x ≤ 7500 

3 75 

37 

x 

5 

≤ 7 500 

3 

75 

x ≤ 20 

You can move at most 20 crates in one trip. 

b. 3 times. Sample answer: Because 20 crates can be 

moved in one trip and 50 crates need to be moved, you 

must divide 50 by 20, which is 2.5. You cannot take 

half of a trip, so the elevator would have to be loaded 

3 times. 

37. Let x the number of pages read each day. 

Number 

of days 

p 

≥ 

7x ≥ 105 

7 x 

≥ 10 5 

 

7 7 

x ≥ 15 

You should read at least 15 pages each day. 

38. d ≤ rt 

45 ≤ r p 5 

4 5 

≤ 5 r 

 

5 5 

9 ≤ r 

An average speed of at least 9 miles per hour will allow 

you to meet your goal. 

39. 8.9 ≥ 40.94 

8.9 

8 

b 

.9 

≤ 4 0.94 

 

8.9 

b ≤ 4.6 

40. 2 

x 

.4 ≥ 8.5 

x 

2.4 p ≥ 2.4 p 8.5 

2 .4 

x ≥ 20.4 

20 20.2 20.4 20.6 20.8 21 

41. 7 

z 

.2 < 3.4 

Number 

of crates 

Number of pages 

read each day 

5 4.8 4.6 4.4 4.2 4 

z 

7.2 p 7 .2 < 7.2(3.4) 

z < 24.48 

Weight 

limit 

Minimum 

number of pages 

24.48 

25 24.8 24.6 24.4 24.2 24 






42. 6.3a > 10.71 

6 .3a 

6.3 

> 10 . 71 

6 

. 3 

a > 1.7 

1 1.2 1.4 1.6 1.8 2 

43. 3.9c ≤ 43.68 

3. 

3 

9c 

. 9 

≥ 4 3.68 

 

3.9 

c ≥ 11.2 

12 11.8 11.6 11.4 11.2 11 

y 

44. ≤ 6.5 9.1 

y 

9.1 p ≥ 9.1 p 6.5 

9.1 

y ≥ 59.15 

60 59.8 59.6 59.4 59.2 59 

45. Let x the number of minutes spent in the shower. 

Gallons per minute Number of Gallons used 

p < 

used in shower minutes in shower in bathtub 

2x < 60 

2 x 

< 6 0 

 

2 2 

x < 30 

You can be in the shower for less than 30 minutes. 

46. a. d ≥ rt 

36 ≥ r p 24 

3 6 

≥ 2 4r 

 

24 

24 

1.5 ≥ r 

Caribou can migrate 1.5 miles per hour or less. 

b. 3 days 72 hours 

72 hours p 1.5 m iles 

108 miles 

hour 

47. A lw 10(12) 120 

The area of your room is 120 square feet. 

Let x the cost per square foot of carpeting. 

Number of 

square feet 

104 106 108 

p 

59.15 

Cost per 

square feet 

≤ 

120x ≤ 200 

110 

Maximum 

amount 

1 20 

12 

x 

0 

≤ 2 0 

1 

0 

20 

x ≤ 1.66 

Your parents will spend a maximum of $1.66 per 

square foot. 

48. Sample answer: 6x < 9 is equivalent to 2x < 3 and 4x < 6. 

49. Let P the water pressure. 

Let d the depth underwater in feet. 

P ≥ 14.7 0.45d 

1500 ≥ 14.7 0.45d 

1500 14.7 ≥ 14.7 0.45d 14.7 

1485.3 ≥ 0.45d 

14 85 

0.4 

.3 

5 

≥ 0 .45d 

0 

.45 

3300 2 3 ≥ d 

The camera can be used at a depth at or above 3300 2 3 feet. 


50. x 3.5 9.2 

x 3.5 3.5 9.2 3.5 

x 5.7 

Check: x 3.5 9.2 

5.7 3.5 9.2 

9.2 9.2 ✓ 

51. x 6.7 5.8 

x 6.7 6.7 5.8 6.7 

x 12.5 

Check: x 6.7 5.8 

12.5 6.7 5.8 

5.8 5.8 ✓ 

52. 44.72 5.2x 

44 . 7 

5. 

2 

2 

5 . 2x 

5 

. 2 

8.6 x 

Check: 44.72 5.2x 

44.72 5.2(8.6) 

44.72 44.72 ✓ 

53. 7 

x 

.6 9.5 

x 

7.6 p 7.6 p 9.5 

7 .6 

x 72.2 

x 

Check: 7 .6 9.5 

7 2. 

2 

 

7. 

6 

9.5 

9.5 9.5 ✓ 



Chapter 3 




54. 3x 5 2x 

3x 5 3x 2x 3x 

5 x 

5 x 

 

1 

1 

5 x 

2x 2(5) 10 

P 4s 4(10) 40 


55. x 12 > 96 

x 12 12 > 96 12 

x > 84 

56. x 17 ≥ 44 

x 17 17 ≥ 44 17 

x ≥ 27 

57. x 26 ≤ 33 

x 26 26 ≤ 33 26 

x ≤ 59 

58. x 14 < 29 

x 14 14 < 29 14 

x < 43 


t 

x 

59. D; ≥ 3 60. I; < 6 

9 

7 

25 

? ≥ 3 

14 

? < 6 

9 

7 

2.7 ≥/ 3 2 < 6 

61. C; 18 ≤ 3p 

18 3p 

≤ 

3 3 

6 ≤ p, orp ≥ 6 


Erika e 

Dawn d 

Matthew m 

1. e 4 

82 83 84 85 86 87 88 

0 9 18 27 36 45 54 

55 56 57 58 59 60 61 

39 40 41 42 43 44 45 

2. c > 4d, c > 4(3), c > 12 

3. e > a, d > a, s > a 

Charlie c 

Anthony a 

Stephanie s 

4. e ≤ 13, c ≤ 13, d ≤ 13, a ≤ 13, m ≤ 13, s ≤ 13 



5. m > 3 cousins 

6. s 6 e 6 4 10 

7. e ≥ 2, c ≥ 2, d ≥ 2, a ≥ 2, m ≥ 2, s ≥ 2 

8. e d 1 

4 d 1 

4 1 d 1 1 

3 d 

9. Either c 6, a 6, or m 6. 

From the information given you know that Erika is 

4 years old, Stephanie is 10 years old, and Dawn is 

3 years old. You are given that Charlie’s age is greater 

than 12, but less than or equal to 13. So, Charlie must be 

13 years old. Dawn is older than Anthony, but Anthony 

must be greater than or equal to 2 years old. So, Anthony 

must be 2 years old. One boy must be 6 years old. 

Because Charlie is 13 years old and Anthony is 2 years 

old, Matthew must be 6 years old. 

In order from least to greatest age, the cousins are: 

Anthony: 2 y, Dawn: 3 y, Erika: 4 y, Matthew: 6 y, 

Stephanie: 10 y, and Charlie: 13 y. 

Lesson 3.6 


1. Let g the average number of goals per game. 

Goals Number Goals 

School 

scored this of games p scored > 

record 

season left per game 

52 12g > 124 

52 12g 52 > 124 52 

12g > 72 

1 2g 

> 7 2 

 

12 

12 

g > 6 

Your team must score, on average, more than 6 goals 

per game. 


1. 5 2x < 20 

5 2x 5 < 20 5 

2x < 15 

2 x 

< 1 5 

 

2 2 

x < 7.5 




2. 5x 12 < 8 

Sample answer: 

(1) Subtract 12 from 5x 12 12 < 8 12 

each side. 

(2) Simplify. 5x < 20 

(3) Divide each side by 5 

 

x 

5 

> 20 

 

5 

5 and reverse the 

inequality symbol. 

(4) Simplify. x > 4 

3. 4x 1 > 1 

4x 1 1 > 1 1 

4x > 0 

3 

4 x 

> 0 4 4 

x > 0 

Check: 4x 1 > 1 

4(2) 1 ? > 1 

9 > 1 ✓ 

4. 7 ≥ 5x 3 

7 3 ≥ 5x 3 3 

10 ≥ 5x 

1 0 

≥ 5 x 

 

5 5 

2 

2 

2 ≥ x, orx ≤ 2 

1 

1 

0 1 2 3 4 

Check: 7 ≥ 5x 3 

7 ? ≥ 5(1) 3 

7 ≥ 2 ✓ 

x 

5. 6 < 14 

2 

0 1 2 3 

x 

6 6 < 14 6 

2 

x 

< 20 2 

x 

2 p > 2(20) 

2 

x > 40 

0 10 20 30 40 50 60 

x 

Check: 6 < 14 

2 

42 

6 ? < 14 

2 

15 < 14 ✓ 



6. 10 > 6 5 

y 

10 6 > 6 5 

y 6 

4 > 5 

y 

5 p 4 > 5 p 5 

y 

20 > y, ory < 20 

Check: 10 > 6 5 

y 

10 ? > 6 1 5 

 

5 

10 > 9 ✓ 

7. 5y 2 ≤ y 34 

5y 2 y ≤ y 34 y 

4y 2 ≤ 34 

4y 2 2 ≤ 34 2 

4y ≤ 32 

4 y 

≤ 3 2 

 

4 4 

y ≤ 8 

Check: 5y 2 ≤ y 34 

5(8) 2 ? ≤ 8 34 

42 ≤ 42 ✓ 

8. 6 y ≥ 2y 3 

6 y y ≥ 2y 3 y 

6 ≥ y 3 

6 3 ≥ y 3 3 

9 ≥ y, ory ≤ 9 

3 

0 5 10 15 20 25 30 

0 2 4 6 8 10 12 

0 3 6 

Check: 6 y ≥ 2y 3 

6 6 ? ≥ 2(6) 3 

12 ≥ 9 ✓ 

9. (1) Let v the number of visits to the park. 

Option 1: Not buying a season pass 

Cost of Number Cost of Number 

p p 

admission of visits parking of visits 

23v 10v 33v 

(2) Option 2: Buying a season pass 

Cost of 

season pass 

 

120 8v 

120 8v < 33v 


Chapter 3 

9 12 15 

Cost of 

parking 

p 

Number 

of visits 





(3) 120 8v 8v < 33v 8v 

120 < 25v 

1 20 

< 2 5v 

 

25 

25 

4.8 < x, orx > 4.8 

Sample answer: After 5 visits to the park, the cost of 

the season pass will be less than the cost of visiting 

without the season pass. 


10. 5x 10 > 2x 4 

5(8) 10 ? > 2(8) 4 

30 > 20 

30 > 20, so 8 is a solution. 

11. 5x 10 > 2x 4 

5(5) 10 ? > 2(5) 4 

15 > 14 

15 > 14, so 5 is a solution. 

12. 5x 10 > 2x 4 

5(4) 10 ? > 2(4) 4 

10 < 12 

10 < 12, so 4 is not a solution. 

13. 5x 10 > 2x 4 

5(2) 10 ? > 2(2) 4 

20 < 0 

20 < 0, so 2 is not a solution. 

14. 2y 7 > 11 

2y 7 7 > 11 7 

2y > 4 

2 y 

> 4 2 2 

y > 2 

1 

0 1 2 

15. 6n 3 ≤ 9 

6n 3 3 ≤ 9 3 

6n ≤ 6 

6 n 

≤ 6 

 

6 6 

n ≤ 1 

4 3 2 1 

16. 11 4z < 1 

11 4z 11 < 1 11 

4z < 12 

4 

 

z 

4 

> 12 

 

4 

z > 3 

3 4 5 

0 1 2 

0 1 2 3 4 5 



6 

17. 3m 8 > 30 5m 

3m 8 5m > 30 5m 5m 

2m 8 > 30 

2m 8 8 > 30 8 

2m > 22 

2 

 

m 

2 

< 22 

 

2 

m < 11 

7 8 9 10 

x 

18. 19 ≥ 25 9 0 

x 

19 25 ≥ 25 25 

9 0 

x 

44 ≥ 

9 0 

x 

90 p 44 ≥ 90 p 

9 0 

3960 ≥ x, orx ≤ 3960 

3880 3920 

19. 3 b 3 < 7 

3 b 3 3 < 7 3 

4 

b 3 < 4 

3 p b 3 < 3 p 4 

b < 12 

0 4 8 

20. 14p 5 ≥ 3p 114 

14p 5 3p ≥ 3p 114 3p 

17p 5 ≥ 114 

17p 5 5 ≥ 114 5 

17p ≥ 119 

1 7p 

≥ 1 19 

 

17 

17 

p ≥ 7 

3 4 5 6 7 8 9 

21. 3x 3 < 2x 83 

3x 3 2x < 2x 83 2x 

5x 3 < 83 

5x 3 3 < 83 3 

5x < 80 

5 

 

x 

5 

> 80 

 

5 

x > 16 

11 12 13 

3960 4000 

12 16 20 

0 4 8 12 16 20 24 




22. Let m the number of movie rentals. 

Option 1: Pay per movie 

Cost of 

movie rental 

p 

4m 

Option 2: One-time membership fee 

Cost of Number 

Membership 

movie p of movie 10 1.50m 

fee 

rental rentals 

10 1.50m < 4m 

10 1.50m 1.50m < 4m 1.50m 

10 < 2.50m 

10 

< 2 .50m 

 

2 .50 

2.50 

4 < m, orm > 4 

The cost of the membership will be less than the 

cost of renting movies without the membership 

after 4 movie rentals. 

23. Let d the number of days the commercial is aired. 

Production Cost Number 

p < Budget 

cost per day of days 

500 50d ≤ 15,000 

500 50d 500 ≤ 15,000 500 

50d ≤ 14,500 

5 0d 

≤ 14 ,500 

 

50 

50 

d ≤ 290 

The company can afford to run the commercial not more 

than 290 days. 

24. Sample answer: In the fourth statement, the direction of 

the inequality symbol should have been reversed because 

each side was divided by a negative number. 

4x > 6x 3 

4x 6x > 6x 3 6x 

2x > 3 

2x 

3 

< 

2 

2 

x < 3 2 

Number of 

movie rentals 

25. 4(5 3b) > 4b 4 

20 12b > 4b 4 

20 12b 4b > 4b 4 4b 

20 16b > 4 

20 16b 20 > 4 20 

16b > 16 

16 

1 

b 

6 

< 1 

 

6 

16 

b < 1 

26. x 2 

> 4 

3 

3 p x 2 

> 3 p 4 

3 

x 2 > 12 

x 2 2 > 12 2 

x > 14 

0 7 14 21 28 35 

27. 3y 5 < 2(17 5y) 

3y 5 < 34 10y 

3y 5 10y < 34 10y 10y 

13y 5 < 34 

13y 5 5 < 34 5 

13y < 39 

1 

1 3y 

< 3 9 

 

13 

13 

y < 3 

28. x 5 

≤ 2 

3 

3 p x 5 

≤ 3 p 2 

3 

x 5 ≤ 6 

x 5 5 ≤ 6 5 

x ≤ 1 

3 

2 

0 1 2 

1 

29. 5s 8 

≥ 22 

4 

4 p 5s 8 

≥ 4(22) 

4 

5s 8 ≥ 88 

5s 8 8 ≥ 88 8 

5s ≥ 80 

5 

 

s 

5 

≤ 80 

 

5 

0 4 8 

s ≤ 16 

12 16 20 

42 

3 4 5 

0 1 2 3 

24 

3 

2 

1 

0 1 2 

3 



Chapter 3 




30. 3 ≤ 2x 4 

 

4 

4(3) ≤ 4 p 2x 4 

 

4 

12 ≤ 2x 4 

12 4 ≤ 2x 4 4 

16 ≤ 2x 

16 2x 

≤ 

2 2 

12 10 

8 ≤ x, orx ≥ 8 

8 

31. Let x the number of sets of cards. 

Cost: Cost per Cards Number Table 

p p 

card in a set of sets cost 

0.50 p 12 p x 20 

6x 20 

Income: Price Number 

p 10.20x 

per set of sets 

10.20x > 6x 20 

10.20x 6x > 6x 20 6x 

4.20x > 20 

4 . 20x 

20 

> 

4. 

20 

4 .20 

x > 4.76 

You must sell at least 5 sets of cards. 

32. a. Let x the number of minutes. 

Monthly 

fee 

6 

 

4 

2 

Per minute 

charge 

Company A: 2 0.039x 

Company B: 0.049x 

2 0.039x < 0.049x 

2 0.039x 0.039x < 0.049x 0.039x 

2 < 0.01x 

2 

0. 01 < 0 . 01x 

 

0. 

01 

200 < x 

The cost of company A will be less than the cost of 

company B after more than 200 minutes. 

b. Company C: 1.95 0.044x 

1.95 0.044x < 0.049x 

1.95 0.044x 0.044x < 0.049x 0.044x 

1.95 < 0.005x 

1.95 

< 0 . 005x 

 

0 .005 

0. 

005 

390 < x 

The cost of company C will be less than the cost of 

company B after more than 390 minutes. 




0 

p 

Number of 

minutes 


c. If you spend 150 minutes each month making 

long-distance calls, use company B. 

Cost using company A: $2 $.039(150) $7.85; 

Cost using company B: $.049(150) $7.35; 

Cost using company C: $1.95 $.044(150) $8.55 

33. x ≥ 4 and x < 3, or all values between 4 and 3, 

including 4, but not 3. 

Sample answer: 

2x 4 < 10 and 5 3x ≤ 17 

2x 4 4 < 10 4 5 3x 5 ≤ 17 5 

2x < 6 3x ≤ 12 

2 x 

< 6 2 2 

3x 

12 

≥ 

3 

3 

x < 3 and x ≥ 4 

Both inequalities are true when x is less than 3 and 

greater than or equal to 4. 

34. a. Let m the number of months that you and your 

friend have been members. 

b. 

Amount paid ($) 

m Amount you Amount your 

months have paid friend has paid 

y 

600 

500 

400 

300 

200 

100 

1 $185 $140 

2 $220 $180 

3 $255 $220 

4 $290 $260 

5 $325 $300 

6 $360 $340 

7 $395 $380 

8 $430 $420 

9 $465 $460 

10 $500 $500 

11 $535 $540 

12 $570 $580 

Health Club Plans 

Me 

My friend 

0 

0 2 4 6 8 10 12m 

Number of months 

c. 11 months. Sample answer: After 10 months, the 

graphs share the same point, which means that the cost 

is the same. Then after that, the points corresponding 

to my cost go below those corresponding to my 

friend’s cost, which means that my plan is cheaper. 






d. Monthly fee p Number of months Membership fee 

Your health club: 150 35m 

Your friend’s health club: 100 40m 

150 35m < 100 40m 

150 35m 40m < 100 40m 40m 

150 5m < 100 

150 5m 150 < 100 150 

5m < 50 

5 

 

m 

5 

> 50 

 

5 

m > 10 

You will have paid less than your friend after more 

than 10 months. 


35. Point A is 1 unit to the right of the origin and 3 units 

up. So, the x-coordinate is 1 and the y-coordinate is 3. 

Point A is represented by the ordered pair (1, 3). 

36. Point B is 3 units to the left of the origin and 1 unit down. 

So, the x-coordinate is 3 and the y-coordinate is 1. 

Point B is represented by the ordered pair (3, 1). 

37. Point C is 3 units to the right of the origin. So, the 

x-coordinate is 3 and the y-coordinate is 0. Point C is 

represented by the ordered pair (3, 0). 

38. Point D is 1 unit to the right of the origin and 3 units 

down. So, the x-coordinate is 1 and the y-coordinate 

is 3. Point D is represented by the ordered pair (1, 3). 

39. Point E is 4 units to the left of the origin and 4 units up. 


Point E is represented by the ordered pair (4, 4). 

40. Point F is 1 unit to the left of the origin and 3 units up. 


Point F is represented by the ordered pair (1, 3). 

41. 13(2a 1) 13(2a) 13(1) 26a 13 

42. 12 c 8 c 12 8 c 20 

43. 5a a (5 1)a 6a 

44. 3(x 4) 9 

3x 12 9 

3x 12 12 9 12 

3x 3 

3 x 

3 

 

3 3 

x 1 

Check: 3(x 4) 9 

3(1 4) 9 

9 9 ✓ 

45. 4(2d 1) 28 

8d 4 28 

8d 4 4 28 4 

8d 24 

8 d 

2 4 

 

8 8 

d 3 

Check: 4(2d 1) 28 

4[2(3) 1] 28 

28 28 ✓ 

46. 10 2(7 2x) 

10 14 4x 

10 14 14 4x 14 

24 4x 

24 

4x 

 

4 

4 

6 x 

Check: 10 2(7 2x) 

10 2[7 2(6)] 

10 10 ✓ 

47. 3x 9 2x 7 

3x 9 2x 2x 7 2x 

x 9 7 

x 9 9 7 9 

x 16 


48. D; 7 6x ≥ 13 

7 6x 7 ≥ 13 7 

6x ≥ 6 

6x 

6 

≤ 

6 

6 

x ≤ 1 

49. I; 7x 3 < 7.5 

7(3) 3 ? < 7.5 

18 < 7.5 

18 < 7.5, so 3 is a solution. 

Chapter 3 Review (pp. 154–157) 

1. The value of a variable that, when substituted into 

an inequality, makes a true statement is a solution of 

an inequality. 

2. Sample answer: 2x 3 ≤ 7 

3. The inequalities of 2x < 2 and x < 1 are equivalent 

inequalities. 



Chapter 3 




4. No. Sample answer: 2x > 6 

2x 

6 

< 

2 

2 

x < 3 

Thus, x < 3 and x > 3 have different solutions. 

So, 2x > 6 and x > 3 are not equivalent inequalities. 

5. Let s the cost of a jar of sauce. 

Total Cost of Jars of Cost of a 

p 

cost spaghetti sauce jar of sauce 

6.49 1.59 2s 

6.49 1.59 1.59 2s 1.59 

4.90 2s 

4. 90 2s 

 

2 2 

2.45 2s 

One jar of sauce costs $2.45. 

6. 17h 47 6h 160 

23h 47 160 

23h 47 47 160 47 

23h 207 

2 3h 

2 07 

 

23 

23 

h 9 

Check: 17h 47 6h 160 

17(9) 47 6(9) 160 

160 160 ✓ 

7. 2(4p 8) 128 

8p 16 128 

8p 16 16 128 16 

8p 112 

8 p 

11 2 

 

8 8 

p 14 

Check: 2(4p 8) 128 

2[4(14) 8] 128 

128 128 ✓ 

8. 6(w 4) 18 30 

6w 24 18 30 

6w 6 30 

6w 6 6 30 6 

6w 36 

6 w 36 

 

6 6 

w 6 

Check: 6(w 4) 18 30 

6(6 4) 18 30 

30 30 ✓ 

9. 11t 14 95 16t 

11t 14 16t 95 16t 16t 

27t 14 95 

27t 14 14 95 14 

27t 81 

2 7t 

8 1 

 

27 

27 

t 3 

Check: 11t 14 95 16t 

11(3) 14 95 16(3) 

47 47 ✓ 

10. 9n 64 144 17n 

9n 64 17n 144 17n 17n 

26n 64 144 

26n 64 64 144 64 

26n 208 

2 6n 

26 

208 

2 

6 

n 8 

Check: 9n 64 144 17n 

9(8) 64 144 17(8) 

8 8 ✓ 

11. 3 2x 2(2 x) 

3 2x 4 2x 

3 2x 2x 4 2x 2x 

3 4 

No solution 

12. 3(2 6b) 18b 

6 18b 18b 

6 18b 18b 18b 18b 

6 0 

No solution 

13. y 11 < 23 

y 11 11 < 23 11 

y < 12 

0 3 6 9 12 15 18 

14. 15 ≥ z 9 

15 9 ≥ z 9 9 

6 ≥ z, orz ≤ 6 

0 2 4 6 8 10 12 

15. x 5 ≤ 14 

x 5 5 ≤ 14 5 

x ≤ 19 

15 16 17 18 19 20 21 






16. m 8 < 26 

m 8 8 < 26 8 

m < 34 

a 

17. 3 > 

9 

a 

9 p 3 < 9 p 

9 

27 < a, ora > 27 

31 30 29 28 27 26 25 

18. b 7 ≥ 13 

7 p b 7 ≥ 7 p 13 

b ≥ 91 

19. 12c ≤ 96 

30 31 32 33 34 35 36 

89 90 91 92 93 94 95 

1 2 

1 

c 

2 

≤ 9 6 

1 

2 

c ≤ 8 

20. 68 < 17d 

68 

> 17d 

 

17 

17 

4 > d, ord < 4 

r 

21. 2 > 

6 

r 

6(2) < 6 p 

6 

12 < r, orr > 12 

22. 196 ≤ 14z 

1 96 

≤ 1 4z 

 

14 

14 

14 ≤ z, orz ≥ 14 

11 12 

23. 7h < 56 

0 2 4 6 8 10 12 

0 1 2 3 4 5 6 

0 4 8 12 16 20 24 

7 h 

< 56 

7 7 

h < 8 

12 10 

13 14 15 16 17 

8 

6 

4 

2 

0 

24. p 5 > 6 

5 p p 5 > 5(6) 

p > 30 

60 50 40 30 20 10 

25. 8m 6 < 10 

8m 6 6 < 10 6 

8m < 16 

8m 

16 

> 

8 

8 

m > 2 

4 3 2 1 

26. 8p 1 ≥ 17 

8p 1 1 ≥ 17 1 

8p ≥ 16 

8 p 

≥ 1 6 

 

8 8 

1 

p ≥ 2 

27. 24 ≥ 5z 6 

24 6 ≥ 5z 6 6 

30 ≥ 5z 

3 0 

≥ 5 z 

 

5 5 

2 

6 ≥ z, orz ≤ 6 

28. 8 > 2 b 3 

8 2 > 2 b 3 2 

6 > b 3 

3 p 6 > 3 p b 3 

18 > b, orb < 18 

6 

p 

29. 3 ≤ 9 2 8 

0 1 2 3 4 5 

0 2 4 6 8 10 

0 6 12 18 24 30 

p 

3 3 ≤ 9 3 

2 8 

p 

≤ 6 2 8 

p 

28 p ≤ 28 p 6 2 8 

p ≤ 168 

0 

0 1 2 



164 165 166 167 168 169 170 

Chapter 3 




30. n 3 4 > 5 

n 3 4 4 > 5 4 

n 3 > 1 

3 p n 3 > 3 p 1 

n > 3 

31. 12 4q ≥ 6q 2 

12 4q 6q ≥ 6q 2 6q 

12 10q ≥ 2 

12 10q 12 ≥ 2 12 

10q ≥ 10 

3 

10 

1 

q 

0 

≤ 1 

 

0 

10 

q ≤ 1 

32. 6x 5 > 12x 1 

6x 5 12x > 12x 1 12x 

6x 5 > 1 

6x 5 5 > 1 5 

6x > 6 

4 

0 1 2 3 4 5 

2 

3 

1 

6x 

6 

< 

6 

6 

x < 1 

2 

1 

0 1 2 3 

0 1 2 

33. 6(3 a) ≤ 8a 10 

18 6a ≤ 8a 10 

18 6a 8a ≤ 8a 10 8a 

18 14a ≤ 10 

18 14a 18 ≤ 10 18 

14a ≤ 28 

14 

1 

a 

4 

≥ 2 

 

8 

14 

a ≥ 2 

1 0 1 2 3 4 5 

34. Let x the number of times you go snowboarding. 

Cost of boots Boots and Number of times 

< p 

and snowboard snowboard rental snowboarding 

360 < 40x 

9 < x 

You must snowboard 10 times or more. 

6 

Chapter 3 Test (p. 158) 

1. 7f 5 68 

7f 5 5 68 5 

7f 63 

7 f 

6 3 

 

7 7 

f 9 

Check: 7f 5 68 

7(9) 5 68 

68 68 ✓ 

2. 14 3g 32 

14 3g 14 32 14 

3g 18 

3g 

18 

 

3 

3 

g 6 

Check: 14 3g 32 

14 3(6) 32 

32 32 ✓ 

3. h 14 11 

3 

h 14 14 11 14 

3 

h 3 3 

3 p h 3 3 p 3 

h 9 

Check: h 14 11 

3 

9 3 14 11 

z 

4. 5 7 

2 

11 11 ✓ 

z 

5 5 7 5 

2 

z 

2 2 

z 

2 p 2 p 2 

2 

z 4 

z 

Check: 5 7 

2 

4 

5 

2 

7 

7 7 ✓ 






5. 12 2m 5 1 

2m 17 1 

2m 17 17 1 17 

2m 18 

2 

 

m 

2 

18 

 

2 

m 9 

Check: 12 2m 5 1 

12 2(9) 5 1 

1 1 ✓ 

6. 6y 4 11y 16 

5y 4 16 

5y 4 4 16 4 

5y 20 

5 y 

20 

5 5 

y 4 

Check: 6y 4 11y 16 

6(4) 4 11(4) 16 

16 16 ✓ 

7. 3(8 a) 12 

24 3a 12 

24 3a 24 12 24 

3a 12 

3 

 

a 

3 

12 

 

3 

a 4 

Check: 3(8 a) 12 

3(8 4) 12 

12 12 ✓ 

8. 6(3x 15) 18 

18x 90 18 

18x 90 90 18 90 

18x 108 

18x 

108 

 

18 

18 

x 6 

Check: 6(3x 15) 18 

5[3(6) 15] 18 

18 18 ✓ 

9. 5t 5 5t 4 

5t 5 5t 5t 4 5t 

5 4 

No solution 

10. 2n 6 8n 14 

2n 6 8n 8n 14 8n 

10n 6 14 

10n 6 6 14 6 

10n 20 

1 0n 

2 0 

 

10 

10 

n 2 

Check: 2n 6 8n 4 

2(2) 6 8(2) 14 

2 2 ✓ 

11. 8b 4 4(b 7) 

8b 4 4b 28 

8b 4 4b 4b 28 4b 

4b 4 28 

4b 4 4 28 4 

4b 32 

4 b 

32 

4 4 

b 8 

Check: 8b 4 4(b 7) 

8(8) 4 4(8 7) 

60 60 ✓ 

12. 16p 8 2(8p 4) 

16p 8 16p 8 


13. Let x the cost of an adult movie ticket. 

Cost of Number Cost of Number 

Total 

child’s p of adult’s p of 

cost 

ticket children ticket adults 

Number of 

Cost of 

p boxes of 

popcorn 

popcorn 

26.50 3.50(2) x(2) 2.50(3) 

26.50 7 2x 7.50 

26.50 2x 14.50 

26.50 14.50 2x 14.50 14.50 

12 2x 

1 2 

2 x 

 

2 2 

6 x 

An adult movie ticket costs $6. 

14. Let x the temperature in C at which ocean 

water freezes. 

x ≤ 1.9 

2 1.8 1.6 1.4 1.2 1 



Chapter 3 




15. x 75 > 125 

x 75 75 > 125 75 

x > 200 

300 200 100 0 

16. w 18 < 10 

w 18 18 < 10 18 

w < 8 

t 

17. ≥ 3 1 2 

t 

12 p ≥ 12 p 3 1 2 

t ≥ 36 

18. 3a 6 ≤ 9 

3a 6 6 ≤ 9 6 

3a ≤ 3 

1 

3 

 

a 

3 

≥ 3 

 

3 

a ≥ 1 

19. 4(2 d) ≥ 12 

8 4d ≥ 12 

8 4d 8 ≥ 12 8 

4d ≥ 20 

4d 

≤ 20 

 

4 

4 

d ≤ 5 

20. 2c 5 < 21 2c 

2c 5 2c < 21 2c 2c 

4c 5 < 21 

4c 5 5 < 21 5 

4c < 16 

6 

0 2 4 6 8 10 12 

0 12 

0 1 2 3 4 5 

0 1 2 3 4 5 

5 

24 36 48 60 72 

4 c 

< 16 

4 4 

4 

c < 4 

3 

2 

1 

6 

0 

21. Let x the cost of a folder. 

Number of folders p Cost of a folder ≤ Total amount 

5x ≤ 5.75 

5 x 

≤ 5. 75 

5 5 

x ≤ 1.15 

You can afford folders that cost $1.15 or less each. 

22. 9 ≥ 15 x 

9 15 ≥ 15 x 15 

6 ≥ x 

6 x 

≤ 

1 

1 

6 ≤ x, orx ≥ 6 

23. 8(5 x) < 56 

40 8x < 56 

40 8x 40 < 56 40 

8x < 16 

8 x 

< 1 6 

 

8 8 

x < 2 

24. 15 > 3(x 4) 

15 > 3x 12 

15 12 > 3x 12 12 

27 > 3x 

2 7 

> 3 x 

 

3 3 

9 > x, orx < 9 

25. 7x 5 ≤ 16 

7x 5 5 ≤ 16 5 

7x ≤ 21 

7 x 

≤ 2 1 

 

7 7 

x ≤ 3 

26. Let x the number of loaves of bread. 

Option 1: Make bread 

Cost of bread Cost of Number 

p 99 0.45x 

machine ingredients of loaves 

Option 2: Buy bread 

Cost of loaf p Number of loaves 2.19x 

99 0.45x < 2.19x 

99 0.45x 0.45x < 2.19x 0.45x 

99 < 1.74x 

99 

< 1 . 74x 

 

1 .74 

1. 

74 

56.9 < x 

You will have to make at least 57 loaves of bread. 






Chapter 3 Standardized Test (p. 159) 

1. D; 5 

t 12 10 

5 

t 12 12 10 12 

5 

t 22 

5 p 5 

t 5 p 22 

t 110 

2. H; 4(n 5) 32 

4n 20 32 

4n 20 20 32 20 

4n 12 

4 

 

n 

4 

12 

 

4 

n 3 

3. C; P a b c 

15 (x 2) 5 (3x 4) 

15 4x 3 

15 3 4x 3 3 

12 4x 

1 2 

4 x 

 

4 4 

3 x 

4. I; 7(s 1) 3 7s 

7s 7 3 7s 

7s 7 7s 3 7s 7s 

7 3 

No solution 

5. C 

z 

6. G; 3 < 15 

4 

z 

3 3 < 15 3 

4 

z 

< 12 4 

z 

4 p > 4 p 12 

4 

z > 48 

7. A; 12 > y 6 

12 6 > y 6 6 

18 > y, or y < 18 

8. I; 5y 2 ≥ 30.5 

5(3) 2 ? ≥ 30.5 

13 ≤ 30.5 

13 ≤ 30.5, so 3 is not a solution. 

9. Let x the cost of one game. 

Cost of Number Cost Number Cost Cost 

bowling of p of of p of of 

shoes games game games game soda 

3 3x 4x 0.50 

3 3x 3x 4x 0.50 3x 

3 x 0.50 

3 0.50 x 0.50 0.50 

2.50 x 

The cost of one game is $2.50 

10. a. Let m the number of miles. 

Daily charge Charge per mile p Number of miles 

Company A: 80 0.35m 

Company B: 75 0.39m 

80 0.35m < 75 0.39m 

80 0.35m 0.35m < 75 0.39m 0.35m 

80 < 75 0.04m 

80 75 < 75 0.04m 75 

5 < 0.04m 

5 

0. 04 < 0 .04m 

 

0.04 

125 < m, or m > 125 

b. 

Unit 1 

0 25 50 75 100 125 150 

c. Company B. Sample answer: 

Company A: 80 0.35(100) 80 35 $115 

Company B: 75 0.39(100) 75 39 $114 

Checkpoint (p. 161) 

1. Answer choice C is unreasonable because 4 people 

paying $24.95 each would make the total payment almost 

$100, but only $29.95 needs to be paid. 

2. Answer choice I is unreasonable because the value of 

5(3.6) 3 is positive, not negative. 

3. Answer choice D is unreasonable because the temperature 

starts at 0F and drops, so it could not be 12F at 1 A.M. 

Test-Taking Practice (pp. 162–163) 

1. A; 12 inches 

3feet 

y yards 36y 

1 ft 

oo 

1 rd ya 

There are 36y inches in y yards. 

2. I; 15 14 2 5 15 7 5 8 5 13 

3. C; 6 (15) 6 15 9C 

4. H 



Chapter 3 




5. C; 3(0.9 7) 3(0.9) 3(7) 

Distributive property 

6. H; 4x and 9x are like terms because they have an identical 

variable part raised to the same power. 

7. A; 10 x 19 

10 x 10 19 10 

x 29 

8. F; 8(x 2) 5(x 3) 8(x) 8(2) 5(x) 5(3) 

8x 16 5x 15 

3x 31 

9. D; 3 8x 141 

3 8x 3 141 3 

8x 144 

8 

 

x 

8 

144 

 

8 

x 18 

10. F; 7(x 5) 10 2x 

7(x) 7(5) 10 2x 

7x 35 10 2x 

7x 25 2x 

7x 25 7x 2x 7x 

25 5x 

25 

5x 

5 5 

5 x 

11. B; 4x 5 < 7 

4x 5 5 < 7 5 

4x < 12 

12. G 

4x 

12 

> 

4 

4 

x > 3 

13. D; 3x 14 > 2x 11 

3x 14 3x > 2x 11 3x 

14 > 5x 11 

14 11 > 5x 11 11 

25 > 5x 

2 5 

> 5 x 

 

5 5 

5 > x, orx < 5 

14. The statement⏐a b⏐⏐a⏐⏐b⏐is sometimes true. For 

example, let a 5 and b 3. 

⏐a b⏐⏐a⏐⏐b⏐ 

⏐5 3⏐ ⏐5⏐⏐3⏐ 

⏐2⏐ 5 3 

2 2 ✓ 





Now let a 5 and b 3. 

⏐a b⏐⏐a⏐⏐b⏐ 

⏐5 (3)⏐ ⏐5⏐⏐3⏐ 

⏐5 3⏐ ⏐5⏐⏐3⏐ 

⏐8⏐ 5 3 

8 2 

15. Addition has both the associative and commutative 

properties, so you can start by adding $4.15 and $1.85. 

Then add $2.74 to get the final total. 

1.85 2.74 4.15 1.85 4.15 2.74 

6.00 2.74 

8.74 

The total cost is $8.74. 

16. Add the profit for each of the 4 months. Then divide by 4 

to find the mean profit. 

670 (340) 320 400 

 

4 

2 90 

72.5 

4 

The mean profit over the 4 months is $72.50. 

17. 4x 45 ≤ 180 

4x 45 45 ≤ 180 45 

4x ≤ 135 

4 x 

≤ 13 5 

 

4 4 

x ≤ 33 3 4 

You can spend at most 33 3 4 minutes, or 33 minutes and 

45 seconds on each of the remaining 4 subjects. 

The inequality is expressed in minutes, where 180 

represents the 3 hours available to do homework and 

(4x 45) represents the time spent doing homework. 

18. a. Total area of walls: 

A 8(16) 8(14) 8(16) 8(14) 

128 112 128 112 

480 

Total area of doors/windows: 

A 3(5) 3(5) 3(7) 3(7) 

15 15 21 21 

72 

Area to paint Area of walls Area of doors/windows 

A 480 72 408 

The total area needing painted is 408 square feet. 

Because you want two coats, you need enough paint to 

cover 2(408) 816 square feet. 

You should buy two 1-gallon cans of paint and one 

1-quart can of paint. 

2(400) 100 800 100 900 

This will paint 900 square feet. 






b. 2(13.90) 8.90 27.80 8.90 36.7 

It will cost $36.70 to put two coats of paint on 

each wall. 

c. Area of walls in closet: 

A 4(8) 4(8) 4(8) 32 32 32 96 

The three walls of the closet have a total area of 

96 square feet. 

Paint left after painting room: 900 816 84 

There is enough paint left to paint 84 square feet, but 

this is not enough to paint the closet. 

There is not enough money left to buy more paint, 

so you cannot afford to paint the inside walls of 

the closet. 

19. a. Four aerobic classes per month is 48 classes per year. 

Gym A: 

Cost Annual fee Class fee p Number of classes 

540 3(48) 

540 144 

684 

Gym A would cost $684. 

Gym B: 

Cost 360 5(48) 360 240 600 

Gym B would cost $600. 

Gym B would be less expensive. 

b. 540 3x < 360 5x 

540 3x 3x < 360 5x 3x 

540 < 360 2x 

540 360 < 360 2x 360 

180 < 2x 

18 0 

< 2 x 

 

2 2 

90 < x 

When the number of aerobics classes is 91 or more, 

gym A costs less than gym B. 

c. Gym B costs less than gym A when x < 90, so you 

should average less than 9 0 

7.5 classes. So, you 

12 should average 7 or fewer aerobics classes per month. 

Cumulative Practice (pp. 164–166) 

1. B; 9 x 9 (5) 4 

2. H 

3. C; 6 4 6 p 6 p 6 p 6 1296 

4. G; 28 7 16 4 16 20 

5. C; 2(x y) 2 2(3 4) 2 2(7) 2 2(49) 98 

6. G; 

5 

3 

0 

4 

8. G; 27 x 27 (8) 27 8 19 

9. D; 12 (8) 12 8 20 

The temperature changed 20C. 

2 

10. F; x ( 4 16 

8 

y 2 

2 

11. C 

) 2 

12. Deposits: 30, 125, 10, 20, 65 

You made deposits totaling $250. 

Withdrawals: 75, 89, 143, 15, 20 

You made withdrawals totaling $342. 

Final balance: 500 250 342 408 

The final balance is $408. 

13. a. To make a scatter plot, graph the ordered pairs from 

the table. Put years on the horizontal axis and 

subscribers on the vertical axis. 

b. 

U.S. Cell Phone Subscribers 

y 

120 

100 

80 

60 

40 

20 

0 

0 1 2 3 4 5 6 x 

Years since 1996 

Subscribers 

(millions) 

c. Yes, the scatter plot suggests that as the number of 

years since 1996 increases, the number of cell phone 

users increases. 

14. C 

15. H; 1.5 miles 52 80 

feet 

7920 feet 

1 me 

il 

16. D; 4x 6 2(2x) 2(3) 2(2x 3) 

17. G; x(y z) 2.5(4 0.1) 2.5(3.9) 9.75 

18. B; 6k and 4k are like terms because they have the same 

variable part raised to the same power. 

19. F; 15y 2(y 3) 15y 2(y) 2(3) 

15y 2y 6 

13y 6 

20. C; d rt 

156 52t 

1 56 

5 2t 

 

52 

52 

3 t 

21. F; x 11 20 7 

x 11 13 

x 11 11 13 11 

x 2 

5 4 3 2 1 0 1 2 

3 4 

7. B; 15 9 6 



Chapter 3 




22. A; 

z 

12 24 

z 

12 p 12 p 24 

12 

z 288 

23. H; Let x amount you spent. 

Beginning amount Amount spent Amount left over 

42.6 x 3.33 

42.6 3.33 x 3.33 3.33 

39.27 x 

24. B; 3y 14.7 

3 

 

y 

3 

1 4. 7 

 

3 

y 4.9 

25. A square has 4 sides of equal lengths. You can express the 

perimeter P as P 4s. 

P 4s 

84 4s 

8 4 

4 s 

 

4 4 

21 s 

The side length of the square is 21 meters. 

26. a. 1500 250 1750 

The original price of the TV at store A is $1750. 

b. Store A offers a better deal. After the rebate and 

delivery, the total price at store A is $1550. Store B 

has no rebate. Their price is 1750 75 $1675. 

27. B; 2x 7 25 

2x 7 7 25 7 

2x 18 

2x 

18 

 

2 

2 

x 9 

28. G; Let x price of breadsticks. 

Total cost Cost of pizza Cost of breadsticks 

12.99 7.99 2x 

12.99 7.99 7.99 2x 7.99 

5 2x 

5 2 2 x 

 

2 

2.5 x 

29. B; 15 2(w 5) 11 

15 2w 10 11 

2w 5 11 

2w 5 5 11 5 

2w 6 

2w 

6 

 

2 

2 

w 3 



30. F; A lw 

28 4(3x 4) 

28 12x 16 

28 16 12x 16 16 

12 12x 

1 2 

1 2x 

 

12 

12 

1 x 

31. D; 2(x 1) 3x (x 2) 

2x 2 3x x 2 

2x 2 2x 2 


32. G 

33. C; Let x the amount you can spend. 

Total 

cost 

≥ 

 

25 ≥ x 13.35 

25 13.35 ≥ x 13.35 13.35 

11.65 ≥ x 

h 

34. H; ≥ 14 7 

h 

7 p ≤ 7 p 14 

7 

h ≤ 98 

35. A; 4s < 42 

4(11) ? < 42 

44 > 42 

So, 11 is not a solution. 

36. F; 4 6x ≥ 8 4x 

4 6x 4x ≥ 8 4x 4x 

4 2x ≥ 8 

4 2x 4 ≥ 8 4 

2x ≥ 12 

2 x 

≥ 12 

2 2 

x ≥ 6 

37. First use the distributive property. Then collect like terms 

and isolate the variable. 

15z 12 3(14 3z) 12 

15z 12 42 9z 12 

15z 12 30 9z 

15z 12 9z 30 9z 9z 

6z 12 30 

6z 12 12 30 12 

6z 42 

6 z 

4 2 

 

6 6 

z 7 

Amount of 

your dinner 

Amount of 

friend’s dinner 




38. Let x number of tickets. 

a. DJ Decoration Cost of Number 

< p 

cost cost tickets of tickets 

125 47.5 < 4.5x 

172.5 < 4.5x 

17 2.5 

4.5 

< 4 . 5x 

4 

. 5 

38 1 3 < x 

You must sell at least 39 tickets for a profit. 

b. DJ Decoration Desired Cost of Number 

≤ p 

cost cost profit tickets of tickets 

125 47.5 300 ≤ 4.5x 

472.5 ≤ 4.5x 

47 2.5 

4.5 

≤ 4 . 5x 

4 

. 5 

105 ≤ x 

You must sell at least 105 tickets for a profit of at 

least $300. 

c. Raising the ticket price to $5.00 will lower the amount 

of tickets needed to be sold in parts (a) and (b). 

Sample answer: The tickets are more expensive, so 

you need to sell fewer tickets to have the same income. 

For part (a), you would start making a profit when 

5x > 125 47.5, or when 35 tickets are sold. For part 

(b), you would start making a profit of at least $300 

when 5x ≥ 125 47.5 300, or when 95 or more 

tickets are sold. 



Chapter 3

Worked-out Problems

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