Practice and Problem-Solving Exercises

A Practice

See Problems 1 and 2.

Solve each equation. Check your answers.

10. | 3 x | = 18 vertical line 3 x vertical line equals 18
11. | − 4 x | = 32 vertical line negative 4 x vertical line equals 32
12. | x − 3 | = 9 vertical line x minus 3 vertical line equals 9
13. 2 | 3 x − 2 | = 14 2 vertical line 3 x minus 2 vertical line equals 14
14. | 3 x + 4 | = − 3 vertical line 3 x plus 4 vertical line equals negative 3
15. | 2 x − 3 | = − 1 vertical line 2 x minus 3 vertical line equals negative 1
16. | x + 4 | + 3 = 17 vertical line x plus 4 vertical line plus 3 equals 17
17. | y − 5 | − 2 = 10 vertical line y minus 5 vertical line negative 2 equals 10
18. | 4 − z | − 10 = 1 vertical line 4 minus z vertical line negative 10 equals 1

See Problem 3.

Solve each equation.Check for extraneous solutions.

19. | x − 1 | = 5 x + 10 vertical line x minus 1 vertical line equals 5 x plus 10
20. | 2 z − 3 | = 4 z − 1 vertical line 2 z minus 3 vertical line equals 4 z minus 1
21. | 3 x + 5 | = 5 x + 2 vertical line 3 x plus 5 vertical line equals 5 x plus 2
22. | 2 y − 4 | = 12 vertical line 2 y minus 4 vertical line equals 12
23. 3 | 4 w − 1 | − 5 = 10 3 vertical line 4 w minus 1 vertical line negative 5 equals 10
24. | 2 x + 5 | = 3 x + 4 vertical line 2 x plus 5 vertical line equals 3 x plus 4

See Problem 4.

Solve each inequality.Graph the solution.

25. 3 | y − 9 | < 27 3 vertical line y minus 9 vertical line less than 27
26. | 6 y − 2 | + 4 < 22 vertical line 6 y minus 2 vertical line plus 4 less than 22
27. | 3 x − 6 | + 3 < 15 vertical line 3 x minus 6 vertical line plus 3 less than 15
28. 1 4 | x − 3 | + 2 < 1 1 fourth , vertical line x minus 3 vertical line plus 2 less than 1
29. 4 | 2 w + 3 | − 7 ≤ 9 4 vertical line 2 w plus 3 vertical line negative 7 less than or equal to 9
30. 3 | 5 t − 1 | + 9 ≤ 23 3 vertical line 5 t minus 1 vertical line plus 9 less than or equal to 23

See Problem 5.

Solve each inequality.Graph the solution.

31. | x + 3 | > 9 vertical line x plus 3 vertical line greater than 9
32. | x − 5 | ≥ 8 vertical line x minus 5 vertical line greater than or equal to 8
33. | y − 3 | ≥ 12 vertical line y minus 3 vertical line greater than or equal to 12
34. | 2 x + 1 | ≥ − 9 vertical line 2 x plus 1 vertical line greater than or equal to negative 9
35. 3 | 2 x − 1 | ≥ 21 3 vertical line 2 x minus 1 vertical line greater than or equal to 21
36. | 3 z | − 4 > 8 vertical line 3 z vertical line negative 4 greater than 8

See Problem 6.

Write each compound inequality as an absolute value inequality.

37. 1.3 ≤ h ≤ 1.5 1.3 less than or equal to h less than or equal to 1.5
38. 50 ≤ k ≤ 51 50 less than or equal to k less than or equal to 51
39. 27.25 ≤ C ≤ 27.75 27.25 , less than or equal to c less than or equal to , 27.75
40. 50 ≤ b ≤ 55 50 less than or equal to b less than or equal to 55
41. 1200 ≤ m ≤ 1300 1200 , less than or equal to m less than or equal to , 1300
42. 0.1187 ≤ d ≤ 0.1190 0.1187 , less than or equal to d less than or equal to , 0.1190

B Apply

Solve each equation.

43. − | 4 − 8 b | = 12 negative vertical line 4 minus 8 b vertical line equals 12
44. 4 | 3 x + 4 | = 4 x + 8 4 vertical line 3 x plus 4 vertical line equals 4 x plus 8
45. | 3 x − 1 | + 10 = 25 vertical line 3 x minus 1 vertical line plus 10 equals 25
46. 1 2 | 3 c + 5 | = 6 c + 4 1 half , vertical line 3 c plus 5 vertical line equals 6 c plus 4
47. 5 | 6 − 5 x | = 15 x − 35 5 vertical line 6 minus 5 x vertical line equals 15 x minus 35
48. 7 | 8 − 3 h | = 21 h − 49 7 vertical line 8 minus 3 h vertical line equals 21 h minus 49
49. 2 | 3 x − 7 | = 10 x − 8 2 vertical line 3 x minus 7 vertical line equals 10 x minus 8
50. 6 | 2 x + 5 | = 6 x + 24 6 vertical line 2 x plus 5 vertical line equals 6 x plus 24
51. 1 4 | 4 x + 7 | = 8 x + 16 1 fourth , vertical line 4 x plus 7 vertical line equals 8 x plus 16
52. 2 3 | 3 x − 6 | = 4 ( x − 2 ) 2 thirds , vertical line 3 x minus 6 vertical line equals 4 open x minus 2 close
53. Think About a Plan The circumference of a basketball for college women must be from 28.5 in. to 29.0 in. What absolute value inequality represents the circumference of the ball?
    • What is the tolerance?
    • What is the inequality without using absolute value?

Write an absolute value equation or inequality to describe each graph.

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