1. Absolute Value
Inequalities

2. What is Absolute Value?

3. What is Absolute Value?
• Absolute Value is the distance from zero on a
number line.

4. What is Absolute Value?
• Absolute Value is the distance from zero on a
number line.
• Distance is always positive.

5. What is Absolute Value?
• Absolute Value is the distance from zero on a
number line.
• Distance is always positive.
• Symbol is | |.

6. What is Absolute Value?
• Absolute Value is the distance from zero on a
number line.
• Distance is always positive.
• Symbol is | |. • Example 1: |3| = 3
-5 -4 -3 -2 -1 0 1 2 3 4 5

7. What is Absolute Value?
• Absolute Value is the distance from zero on a
number line.
• Distance is always positive.
• Symbol is | |. • Example 1: |3| = 3
-5 -4 -3 -2 -1 0 1 2 3 4 5

8. What is Absolute Value?
• Absolute Value is the distance from zero on a
number line.
• Distance is always positive.
• Symbol is | |. • Example 1: |3| = 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Example 2: |-3| = 3
-5 -4 -3 -2 -1 0 1 2 3 4 5

9. What is Absolute Value?
• Absolute Value is the distance from zero on a
number line.
• Distance is always positive.
• Symbol is | |. • Example 1: |3| = 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Example 2: |-3| = 3
-5 -4 -3 -2 -1 0 1 2 3 4 5

10. Absolute Value Equation

11. Absolute Value Equation
• Because the distance can be towards the
positive or negative direction, Absolute Value
Equations must account for both cases.

12. Absolute Value Equation
• Because the distance can be towards the
positive or negative direction, Absolute Value
Equations must account for both cases.
• Example 3: |x| = 4

13. Absolute Value Equation
• Because the distance can be towards the
positive or negative direction, Absolute Value
Equations must account for both cases.
• Example 3: |x| = 4
x = 4 or -5 -4 -3 -2 -1 0 1 2 3 4 5

14. Absolute Value Equation
• Because the distance can be towards the
positive or negative direction, Absolute Value
Equations must account for both cases.
• Example 3: |x| = 4
x = 4 or -5 -4 -3 -2 -1 0 1 2 3 4 5
-(x) = 4

15. Absolute Value Equation
• Because the distance can be towards the
positive or negative direction, Absolute Value
Equations must account for both cases.
• Example 3: |x| = 4
x = 4 or -5 -4 -3 -2 -1 0 1 2 3 4 5
-(x) = 4
x = -4
-5 -4 -3 -2 -1 0 1 2 3 4 5

16. Absolute Value Inequality

17. Absolute Value Inequality
• How does Absolute Value work for Inequalities?

18. Absolute Value Inequality
• How does Absolute Value work for Inequalities?
• Example 4: |x| < 4

19. Absolute Value Inequality
• How does Absolute Value work for Inequalities?
• Example 4: |x| < 4
Write the inequalities for the 2 cases and solve.

20. Absolute Value Inequality
• How does Absolute Value work for Inequalities?
• Example 4: |x| < 4
Write the inequalities for the 2 cases and solve.
Positive Case

21. Absolute Value Inequality
• How does Absolute Value work for Inequalities?
• Example 4: |x| < 4
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case

22. Absolute Value Inequality
• How does Absolute Value work for Inequalities?
• Example 4: |x| < 4
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x<4 -(x) < 4

23. Absolute Value Inequality
• How does Absolute Value work for Inequalities?
• Example 4: |x| < 4
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x<4 -(x) < 4
x > -4

24. Absolute Value Inequality
• How does Absolute Value work for Inequalities?
• Example 4: |x| < 4
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x<4 -(x) < 4
x > -4
Remember to reverse the inequality when
dividing by a negative.

25. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5

26. Example 4 (con’t)
|x| < 4
• Graph the solutions.
-5 -4 -3 -2 -1 0 1 2 3 4 5

27. Example 4 (con’t)
|x| < 4
• Graph the solutions.
x<4
-5 -4 -3 -2 -1 0 1 2 3 4 5

28. Example 4 (con’t)
|x| < 4
• Graph the solutions.
x<4 x > -4
-5 -4 -3 -2 -1 0 1 2 3 4 5

29. Example 4 (con’t)
|x| < 4
• Graph the solutions.
x<4 x > -4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Is the solution is an intersection (∩) or a union (∪)?

30. Example 4 (con’t)
|x| < 4
• Graph the solutions.
x<4 x > -4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Is the solution is an intersection (∩) or a union (∪)?
• Check numbers in the original inequality to help
you decide.

31. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5

32. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x < 4, one that solves x
> -4, and one that solves both inequalities.

33. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x < 4, one that solves x
> -4, and one that solves both inequalities.
• Check 0 first. Why zero? Because it’s easy. ☺

34. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x < 4, one that solves x
> -4, and one that solves both inequalities.
• Check 0 first. Why zero? Because it’s easy. ☺
|0| < 4 True, so the area between -4 and 4 is a solution.

35. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x < 4, one that solves x
> -4, and one that solves both inequalities.
• Check 0 first. Why zero? Because it’s easy. ☺
|0| < 4 True, so the area between -4 and 4 is a solution.
• Check -5.

36. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x < 4, one that solves x
> -4, and one that solves both inequalities.
• Check 0 first. Why zero? Because it’s easy. ☺
|0| < 4 True, so the area between -4 and 4 is a solution.
• Check -5.
|-5| < 4 False, so the area less than -4 is not a solution.

37. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x < 4, one that solves x
> -4, and one that solves both inequalities.
• Check 0 first. Why zero? Because it’s easy. ☺
|0| < 4 True, so the area between -4 and 4 is a solution.
• Check -5.
|-5| < 4 False, so the area less than -4 is not a solution.
• Check 5.

38. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x < 4, one that solves x
> -4, and one that solves both inequalities.
• Check 0 first. Why zero? Because it’s easy. ☺
|0| < 4 True, so the area between -4 and 4 is a solution.
• Check -5.
|-5| < 4 False, so the area less than -4 is not a solution.
• Check 5.
|5| < 4 False, so the area greater than 4 is not a solution.

39. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Because the only number that worked was between
-4 and 4, the solution is an intersection.

40. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Because the only number that worked was between
-4 and 4, the solution is an intersection.
• The solution can be written as

41. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Because the only number that worked was between
-4 and 4, the solution is an intersection.
• The solution can be written as
{x | -4 < x < 4}

42. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Because the only number that worked was between
-4 and 4, the solution is an intersection.
• The solution can be written as
{x | -4 < x < 4}
or as

43. Example 4 (con’t)
|x| < 4
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Because the only number that worked was between
-4 and 4, the solution is an intersection.
• The solution can be written as
{x | -4 < x < 4}
or as
{x | x > -4 ∩ x < 4}

44. Example 5: What about |x| > 3?

45. Example 5: What about |x| > 3?
Write the inequalities for the 2 cases and solve.

46. Example 5: What about |x| > 3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case

47. Example 5: What about |x| > 3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x>3 -(x) > 3

48. Example 5: What about |x| > 3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x>3 -(x) > 3
x < -3

49. Example 5: What about |x| > 3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x>3 -(x) > 3
x < -3
Remember to reverse the inequality when
dividing by a negative.

50. Example 5: What about |x| > 3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x>3 -(x) > 3
x < -3
Remember to reverse the inequality when
dividing by a negative.
• Graph.
-5 -4 -3 -2 -1 0 1 2 3 4 5

51. Example 5: What about |x| > 3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x>3 -(x) > 3
x < -3
Remember to reverse the inequality when
dividing by a negative.
• Graph.
-5 -4 -3 -2 -1 0 1 2 3 4 5

52. Example 5: What about |x| > 3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x>3 -(x) > 3
x < -3
Remember to reverse the inequality when
dividing by a negative.
• Graph.
-5 -4 -3 -2 -1 0 1 2 3 4 5

53. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5

54. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x > 3, one that solves x < 3,
and one that neither inequality.

55. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x > 3, one that solves x < 3,
and one that neither inequality.
• Check 0 first. Why zero? Because it’s easy. ☺

56. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x > 3, one that solves x < 3,
and one that neither inequality.
• Check 0 first. Why zero? Because it’s easy. ☺
|0| > 3 False, so the area between -4 and 4 is not a solution.

57. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x > 3, one that solves x < 3,
and one that neither inequality.
• Check 0 first. Why zero? Because it’s easy. ☺
|0| > 3 False, so the area between -4 and 4 is not a solution.
• Check -5.

58. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x > 3, one that solves x < 3,
and one that neither inequality.
• Check 0 first. Why zero? Because it’s easy. ☺
|0| > 3 False, so the area between -4 and 4 is not a solution.
• Check -5.
|-5| > 3 True, so the area less than -4 is a solution.

59. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x > 3, one that solves x < 3,
and one that neither inequality.
• Check 0 first. Why zero? Because it’s easy. ☺
|0| > 3 False, so the area between -4 and 4 is not a solution.
• Check -5.
|-5| > 3 True, so the area less than -4 is a solution.
• Check 5.

60. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Check 3 numbers. One that solves x > 3, one that solves x < 3,
and one that neither inequality.
• Check 0 first. Why zero? Because it’s easy. ☺
|0| > 3 False, so the area between -4 and 4 is not a solution.
• Check -5.
|-5| > 3 True, so the area less than -4 is a solution.
• Check 5.
|5| > 3 True, so the area greater than 4 is a solution.

61. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5

62. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Because a number worked from both
inequalities, the solution is an union.

63. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Because a number worked from both
inequalities, the solution is an union.
• The solution can be written as

64. Example 5 (con’t): |x| > 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• Because a number worked from both
inequalities, the solution is an union.
• The solution can be written as
{x | x < -3 ∪ x > 3}

65. Example 6: What about |x| < -3?

66. Example 6: What about |x| < -3?
Write the inequalities for the 2 cases and solve.

67. Example 6: What about |x| < -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case

68. Example 6: What about |x| < -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x < -3 -(x) < -3

69. Example 6: What about |x| < -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x < -3 -(x) < -3
x>3

70. Example 6: What about |x| < -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x < -3 -(x) < -3
x>3
• Graph.
-5 -4 -3 -2 -1 0 1 2 3 4 5

71. Example 6: What about |x| < -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x < -3 -(x) < -3
x>3
• Graph.
-5 -4 -3 -2 -1 0 1 2 3 4 5

72. Example 6: What about |x| < -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x < -3 -(x) < -3
x>3
• Graph.
-5 -4 -3 -2 -1 0 1 2 3 4 5

73. Example 6 (con’t): |x| < -3
-5 -4 -3 -2 -1 0 1 2 3 4 5

74. Example 6 (con’t): |x| < -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• It looks a lot like Example 5 so far.
Time for our check. Let’s check -4 first.

75. Example 6 (con’t): |x| < -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• It looks a lot like Example 5 so far.
Time for our check. Let’s check -4 first. ?
− ( −4 ) <− 3
4 < −3
4 < −3

76. Example 6 (con’t): |x| < -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• It looks a lot like Example 5 so far.
Time for our check. Let’s check -4 first. ?
− ( −4 ) <− 3
• What?! This isn’t true! 4 < −3
4 < −3

77. Example 6 (con’t): |x| < -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• It looks a lot like Example 5 so far.
Time for our check. Let’s check -4 first. ?
− ( −4 ) <− 3
• What?! This isn’t true! 4 < −3
• Absolute Value is distance. Distance is 4 < −3
never negative. The Absolute Value
can never result in a negative number.

78. Example 6 (con’t): |x| < -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• It looks a lot like Example 5 so far.
Time for our check. Let’s check -4 first. ?
− ( −4 ) <− 3
• What?! This isn’t true! 4 < −3
• Absolute Value is distance. Distance is 4 < −3
never negative. The Absolute Value
can never result in a negative number. No Solution

79. Example 7: What about |x| > -3?

80. Example 7: What about |x| > -3?
Write the inequalities for the 2 cases and solve.

81. Example 7: What about |x| > -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case

82. Example 7: What about |x| > -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x > -3 -(x) > -3

83. Example 7: What about |x| > -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x > -3 -(x) > -3
x<3

84. Example 7: What about |x| > -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x > -3 -(x) > -3
x<3
• Graph.
-5 -4 -3 -2 -1 0 1 2 3 4 5

85. Example 7: What about |x| > -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x > -3 -(x) > -3
x<3
• Graph.
-5 -4 -3 -2 -1 0 1 2 3 4 5

86. Example 7: What about |x| > -3?
Write the inequalities for the 2 cases and solve.
Positive Case Negative Case
x > -3 -(x) > -3
x<3
• Graph.
-5 -4 -3 -2 -1 0 1 2 3 4 5

87. Example 7 (con’t): |x| > -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• What is different about the graph?

88. Example 7 (con’t): |x| > -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• What is different about the graph?
• Check 0.

89. Example 7 (con’t): |x| > -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• What is different about the graph? ?
0 >− 3
• Check 0.
0 > −3

90. Example 7 (con’t): |x| > -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• What is different about the graph? ?
0 >− 3
• Check 0.
0 > −3
• Looks good. Try -5.

91. Example 7 (con’t): |x| > -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• What is different about the graph? ?
0 >− 3
• Check 0.
0 > −3 ?
−5 >− 3
• Looks good. Try -5.
5 > −3

92. Example 7 (con’t): |x| > -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• What is different about the graph? ?
0 >− 3
• Check 0.
0 > −3 ?
−5 >− 3
• Looks good. Try -5.
5 > −3
• Also looks good.

93. Example 7 (con’t): |x| > -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• What is different about the graph? ?
0 >− 3
• Check 0.
0 > −3 ?
−5 >− 3
• Looks good. Try -5.
5 > −3
• Also looks good.
• Is there a number that will not work in the original problem?

94. Example 7 (con’t): |x| > -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• What is different about the graph? ?
0 >− 3
• Check 0.
0 > −3 ?
−5 >− 3
• Looks good. Try -5.
5 > −3
• Also looks good.
• Is there a number that will not work in the original problem?
• No, which means the solution can be any real number.

95. Example 7 (con’t): |x| > -3
-5 -4 -3 -2 -1 0 1 2 3 4 5
• What is different about the graph? ?
0 >− 3
• Check 0.
0 > −3 ?
−5 >− 3
• Looks good. Try -5.
5 > −3
• Also looks good.
• Is there a number that will not work in the original problem?
• No, which means the solution can be any real number.
All Real Numbers