Section Exercises

1. How do you solve an absolute value equation? 2. How can you tell whether an absolute value function has two x-intercepts without graphing the function? 3. When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function? 4. How can you use the graph of an absolute value function to determine the x-values for which the function values are negative? 5. How do you solve an absolute value inequality algebraically? 6. Describe all numbers [latex]x[/latex] that are at a distance of 4 from the number 8. Express this using absolute value notation. 7. Describe all numbers [latex]x[/latex] that are at a distance of [latex]\frac{1}{2}[/latex] from the number −4. Express this using absolute value notation. 8. Describe the situation in which the distance that point [latex]x[/latex] is from 10 is at least 15 units. Express this using absolute value notation. 9. Find all function values [latex]f\left(x\right)[/latex] such that the distance from [latex]f\left(x\right)[/latex] to the value 8 is less than 0.03 units. Express this using absolute value notation. For the following exercises, solve the equations below and express the answer using set notation.

10. [latex]|x+3|=9[/latex]

11. [latex]|6-x|=5[/latex]

12. [latex]|5x - 2|=11[/latex]

13. [latex]|4x - 2|=11[/latex]

14. [latex]2|4-x|=7[/latex]

15. [latex]3|5-x|=5[/latex]

16. [latex]3|x+1|-4=5[/latex]

17. [latex]5|x - 4|-7=2[/latex]

18. [latex]0=-|x - 3|+2[/latex]

19. [latex]2|x - 3|+1=2[/latex]

20. [latex]|3x - 2|=7[/latex]

21. [latex]|3x - 2|=-7[/latex]

22. [latex]\left|\frac{1}{2}x - 5\right|=11[/latex]

23. [latex]\left|\frac{1}{3}x+5\right|=14[/latex]

24. [latex]-\left|\frac{1}{3}x+5\right|+14=0[/latex]

For the following exercises, find the x- and y-intercepts of the graphs of each function.

25. [latex]f\left(x\right)=2|x+1|-10[/latex]

26. [latex]f\left(x\right)=4|x - 3|+4[/latex]

27. [latex]f\left(x\right)=-3|x - 2|-1[/latex]

28. [latex]f\left(x\right)=-2|x+1|+6[/latex]

For the following exercises, solve each inequality and write the solution in interval notation.

29. [latex]\left|x - 2\right|>10[/latex]

30. [latex]2|v - 7|-4\ge 42[/latex]

31. [latex]|3x - 4|\le 8[/latex]

32. [latex]|x - 4|\ge 8[/latex]

33. [latex]|3x - 5|\ge 13[/latex]

34. [latex]|3x - 5|\ge -13[/latex]

35. [latex]\left|\frac{3}{4}x - 5\right|\ge 7[/latex]

36. [latex]\left|\frac{3}{4}x - 5\right|+1\le 16[/latex]

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

37. [latex]y=|x - 1|[/latex]

38. [latex]y=|x+1|[/latex]

39. [latex]y=|x|+1[/latex]

For the following exercises, graph the given functions by hand.

40. [latex]y=|x|-2[/latex]

41. [latex]y=-|x|[/latex]

42. [latex]y=-|x|-2[/latex]

43. [latex]y=-|x - 3|-2[/latex]

44. [latex]f\left(x\right)=-|x - 1|-2[/latex]

45. [latex]f\left(x\right)=-|x+3|+4[/latex]

46. [latex]f\left(x\right)=2|x+3|+1[/latex]

47. [latex]f\left(x\right)=3|x - 2|+3[/latex]

48. [latex]f\left(x\right)=|2x - 4|-3[/latex]

49. [latex]f\left(x\right)=|3x+9|+2[/latex]

50. [latex]f\left(x\right)=-|x - 1|-3[/latex]

51. [latex]f\left(x\right)=-|x+4|-3[/latex]

52. [latex]f\left(x\right)=\frac{1}{2}\left|x+4\right|-3[/latex]

53. Use a graphing utility to graph [latex]f\left(x\right)=10|x - 2|[/latex] on the viewing window [latex]\left[0,4\right][/latex]. Identify the corresponding range. Show the graph.

54. Use a graphing utility to graph [latex]f\left(x\right)=-100|x|+100[/latex] on the viewing window [latex]\left[-5,5\right][/latex]. Identify the corresponding range. Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

55. [latex]f\left(x\right)=\left(-0.1\right)\left|0.1\left(0.2-x\right)\right|+0.3[/latex]

56. [latex]f\left(x\right)=4\times {10}^{9}\left|x-\left(5\times {10}^{9}\right)\right|+2\times {10}^{9}[/latex]

For the following exercises, solve the inequality.

57. [latex]\left|-2x-\frac{2}{3}\left(x+1\right)\right|+3>-1[/latex]

58. If possible, find all values of [latex]a[/latex] such that there are no [latex]x\text{-}[/latex] intercepts for [latex]f\left(x\right)=2|x+1|+a[/latex].

59. If possible, find all values of [latex]a[/latex] such that there are no [latex]y[/latex] -intercepts for [latex]f\left(x\right)=2|x+1|+a[/latex].

60. Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and [latex]x[/latex] represents the distance from city B to city A, express this using absolute value notation.

61. The true proportion [latex]p[/latex] of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

62. Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable [latex]x[/latex] for the score.

63. A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using [latex]x[/latex] as the diameter of the bearing, write this statement using absolute value notation.

64. The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is [latex]x[/latex] inches, express the tolerance using absolute value notation.

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