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3.5 Solving Equations and Inequalities with Absolute Value

Solve equations with absolute value.
Solve inequalities with absolute value.

Equations with Absolute Value

Recall that the absolute value of a number is its distance from 0 on the number line. We use this concept to solve equations with absolute value.

Example 1

Solve: $| x | = 5$ $| x | = 5$ .

Now Try Exercise 1.

Example 2

Solve: $| x - 3 | - 1 = 4$ $| x - 3 | - 1 = 4$ .

Solution

First, we add 1 on both sides to get an expression of the form $| X | = a$ $| X | = a$ :

\begin{array}{l} \begin{array}{l} | x - 3 | - 1 & = & 4 \\ | x - 3 | & = & 5 \end{array} \\ \begin{array}{rclcrcll} x - 3 & = & −5 & or & x - 3 & = & 5 & | X | = a is equivalent \\ to X = - a or X = a . \\ x & = & −2 & or & x & = & 8. & Adding 3 \end{array} \end{array}

$\begin{array}{l} \begin{array}{l} | x - 3 | - 1 & = & 4 \\ | x - 3 | & = & 5 \end{array} \\ \begin{array}{rclcrcll} x - 3 & = & −5 & or & x - 3 & = & 5 & | X | = a is equivalent \\ to X = - a or X = a . \\ x & = & −2 & or & x & = & 8. & Adding 3 \end{array} \end{array}$

Check:

For −2:

For 8:

The solutions are −2 and 8.

Now Try Exercise 21.

When $a = 0, | X | = a$ $a = 0, | X | = a$ is equivalent to $X = 0$ $X = 0$ . Note that for $a < 0, | X | = a$ $a < 0, | X | = a$ has no solution, because the absolute value of an expression is never negative. We can use a graph to illustrate the last statement for a specific value of a. For example, if we let $a = −3$ $a = −3$ and graph $y = | x |$ $y = | x |$ and $y = −3$ $y = −3$ , we see that the graphs do not intersect, as shown below. Thus the equation $| x | = −3$ $| x | = −3$ has no solution. The solution set is the empty set, denoted $\emptyset$ $\emptyset$ .

Inequalities with Absolute Value

Inequalities sometimes contain absolute-value notation. The following properties are used to solve them.

For example,

$| x | < 3 is equivalent to −3 < x < 3$ $| x | < 3 is equivalent to −3 < x < 3$ ;
$| y | \geq 1 is equivalent to y \leq −1 or y \geq 1; and$ $| y | \geq 1 is equivalent to y \leq −1 or y \geq 1; and$
$| 2 x + 3 | \leq 4 is equivalent to −4 \leq 2 x + 3 \leq 4$ $| 2 x + 3 | \leq 4 is equivalent to −4 \leq 2 x + 3 \leq 4$ .

Example 3

Solve and graph the solution set: $| 3 x + 2 | < 5$ $| 3 x + 2 | < 5$ .

Solution

We have

\begin{array}{rcl} | 3 x + 2 | & < & 5 \\ −5 & < & 3 x + 2 < 5 & Writing an equivalent inequality \\ −7 & < & 3 x < 3 & Subtracting 2 \\ - \frac{7}{3} & < & x < 1. & Dividing by 3 \end{array}

$\begin{array}{rcl} | 3 x + 2 | & < & 5 \\ −5 & < & 3 x + 2 < 5 & Writing an equivalent inequality \\ −7 & < & 3 x < 3 & Subtracting 2 \\ - \frac{7}{3} & < & x < 1. & Dividing by 3 \end{array}$

The solution set is ${x | - \frac{7}{3} < x < 1}$ ${x | - \frac{7}{3} < x < 1}$ , or $(- \frac{7}{3}, 1)$ $(- \frac{7}{3}, 1)$ . The graph of the solution set is shown below.

Now Try Exercise 45.

Example 4

Solve and graph the solution set: $| 5 - 2 x | \geq 1$ $| 5 - 2 x | \geq 1$ .

Solution

We have

\begin{array}{l} | 5 - 2 x | \geq 1 \\ \begin{array}{rclcrcll} 5 - 2 x & \leq & −1 & or & 5 - 2 x & \geq & 1 & Writing an equivalent inequality \\ −2 x & \leq & −6 & or & −2 x & \geq & −4 & Subtracting 5 \\ x & \geq & 3 & or & x & \leq & 2. & Dividing by - 2 and reversing \\ the inequality signs \end{array} \end{array}

$\begin{array}{l} | 5 - 2 x | \geq 1 \\ \begin{array}{rclcrcll} 5 - 2 x & \leq & −1 & or & 5 - 2 x & \geq & 1 & Writing an equivalent inequality \\ −2 x & \leq & −6 & or & −2 x & \geq & −4 & Subtracting 5 \\ x & \geq & 3 & or & x & \leq & 2. & Dividing by - 2 and reversing \\ the inequality signs \end{array} \end{array}$

The solution set is ${x | x \leq 2 or x \geq 3}$ ${x | x \leq 2 or x \geq 3}$ , or $(- \infty, 2] \cup [3, \infty)$ $(- \infty, 2] \cup [3, \infty)$ . The graph of the solution set is shown below.

Now Try Exercise 47.

3.5 Exercise Set

Solve.

1. $| x | = 7$ $| x | = 7$
2. $| x | = 4.5$ $| x | = 4.5$
3. $| x | = 0$ $| x | = 0$
4. $| x | = \frac{3}{2}$ $| x | = \frac{3}{2}$
5. $| x | = \frac{5}{6}$ $| x | = \frac{5}{6}$
6. $| x | = - \frac{3}{5}$ $| x | = - \frac{3}{5}$
7. $| x | = −10.7$ $| x | = −10.7$
8. $| x | = 12$ $| x | = 12$
9. $| 3 x | = 1$ $| 3 x | = 1$
10. $| 5 x | = 4$ $| 5 x | = 4$
11. $| 8 x | = 24$ $| 8 x | = 24$
12. $| 6 x | = 0$ $| 6 x | = 0$
13. $| x - 1 | = 4$ $| x - 1 | = 4$
14. $| x - 7 | = 5$ $| x - 7 | = 5$
15. $| x + 2 | = 6$ $| x + 2 | = 6$
16. $| x + 5 | = 1$ $| x + 5 | = 1$
17. $| 3 x + 2 | = 1$ $| 3 x + 2 | = 1$
18. $| 7 x - 4 | = 8$ $| 7 x - 4 | = 8$
19. $| \frac{1}{2} x - 5 | = 17$ $| \frac{1}{2} x - 5 | = 17$
20. $| \frac{1}{3} x - 4 | = 13$ $| \frac{1}{3} x - 4 | = 13$
21. $| x - 1 | + 3 = 6$ $| x - 1 | + 3 = 6$
22. $| x + 2 | −5 = 9$ $| x + 2 | −5 = 9$
23. $| x + 3 | −2 = 8$ $| x + 3 | −2 = 8$
24. $| x - 4 | + 3 = 9$ $| x - 4 | + 3 = 9$
25. $| 3 x + 1 | −4 = −1$ $| 3 x + 1 | −4 = −1$
26. $| 2 x - 1 | −5 = −3$ $| 2 x - 1 | −5 = −3$
27. $| 4 x - 3 | + 1 = 7$ $| 4 x - 3 | + 1 = 7$
28. $| 5 x + 4 | + 2 = 5$ $| 5 x + 4 | + 2 = 5$
29. $12 - | x + 6 | = 5$ $12 - | x + 6 | = 5$
30. $9 - | x - 2 | = 7$ $9 - | x - 2 | = 7$
31. $7 - | 2 x - 1 | = 6$ $7 - | 2 x - 1 | = 6$
32. $5 - | 4 x + 3 | = 2$ $5 - | 4 x + 3 | = 2$

Solve and write interval notation for the solution set. Then graph the solution set.

33. $| x | < 7$ $| x | < 7$
34. $| x | \leq 4.5$ $| x | \leq 4.5$
35. $| x | \leq 2$ $| x | \leq 2$
36. $| x | < 3$ $| x | < 3$
37. $| x | \geq 4.5$ $| x | \geq 4.5$
38. $| x | > 7$ $| x | > 7$
39. $| x | > 3$ $| x | > 3$
40. $| x | \geq 2$ $| x | \geq 2$
41. $| 3 x | < 1$ $| 3 x | < 1$
42. $| 5 x | \leq 4$ $| 5 x | \leq 4$
43. $| 2 x | \geq 6$ $| 2 x | \geq 6$
44. $| 4 x | > 20$ $| 4 x | > 20$
45. $| x + 8 | < 9$ $| x + 8 | < 9$
46. $| x + 6 | \leq 10$ $| x + 6 | \leq 10$
47. $| x + 8 | \geq 9$ $| x + 8 | \geq 9$
48. $| x + 6 | > 10$ $| x + 6 | > 10$
49. $| x - \frac{1}{4} | < \frac{1}{2}$ $| x - \frac{1}{4} | < \frac{1}{2}$
50. $| x −0.5 | \leq 0.2$ $| x −0.5 | \leq 0.2$
51. $| 2 x + 3 | \leq 9$ $| 2 x + 3 | \leq 9$
52. $| 3 x + 4 | < 13$ $| 3 x + 4 | < 13$
53. $| x - 5 | > 0.1$ $| x - 5 | > 0.1$
54. $| x - 7 | \geq 0.4$ $| x - 7 | \geq 0.4$
55. $| 6 - 4 x | \geq 8$ $| 6 - 4 x | \geq 8$
56. $| 5 - 2 x | > 10$ $| 5 - 2 x | > 10$
57. $| x + \frac{2}{3} | \leq \frac{5}{3}$ $| x + \frac{2}{3} | \leq \frac{5}{3}$
58. $| x + \frac{3}{4} | < \frac{1}{4}$ $| x + \frac{3}{4} | < \frac{1}{4}$
59. $| \frac{2 x + 1}{3} | > 5$ $| \frac{2 x + 1}{3} | > 5$
60. $| \frac{2 x - 1}{3} | \geq \frac{5}{6}$ $| \frac{2 x - 1}{3} | \geq \frac{5}{6}$
61. $| 2 x - 4 | < −5$ $| 2 x - 4 | < −5$
62. $| 3 x + 5 | < 0$ $| 3 x + 5 | < 0$
63. $| 7 - x | \geq −4$ $| 7 - x | \geq −4$
64. $| 2 x + 1 | > - \frac{1}{2}$ $| 2 x + 1 | > - \frac{1}{2}$

Skill Maintenance

Vocabulary Reinforcement

In each of Exercises 65–72, fill in the blank with the correct term. Some of the given choices will not be used.

distance formula
midpoint formula
function
relation
x-intercept
y-intercept
perpendicular
parallel
horizontal lines
vertical lines
symmetric with respect to the x-axis
symmetric with respect to the y-axis
symmetric with respect to the origin
increasing
decreasing
constant

65. A(n) _______________ is a point (0, b). [1.1]
66. The _______________ is $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ . [1.1]
67. A(n) _______________ is a correspondence such that each member of the domain corresponds to at least one member of the range. [1.2]
68. A(n) _______________ is a correspondence such that each member of the domain corresponds to exactly one member of the range. [1.2]
69. _______________ are given by equations of the type $y = b$ $y = b$ , or $f (x) = b$ $f (x) = b$ . [1.3]
70. Nonvertical lines are _______________ if and only if they have the same slope and different y-intercepts. [1.4]
71. A function f is said to be _______________ on an open interval I if, for all a and b in that interval, $a < b$ $a < b$ implies $f (a) > f (b)$ $f (a) > f (b)$ . [2.1]
72. For an equation $y = f (x)$ $y = f (x)$ , if replacing x with −x produces an equivalent equation, then the graph is _______________. [2.4]

Synthesis

Solve.

73. $| 3 x - 1 | > 5 x - 2$ $| 3 x - 1 | > 5 x - 2$
74. $| x + 2 | \leq | x - 5 |$ $| x + 2 | \leq | x - 5 |$
75. $| p - 4 | + | p + 4 | < 8$ $| p - 4 | + | p + 4 | < 8$
76. $| x | + | x + 1 | < 10$ $| x | + | x + 1 | < 10$
77. $| x - 3 | + | 2 x + 5 | > 6$ $| x - 3 | + | 2 x + 5 | > 6$

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Table of Contents for 3.5 Solving Equations and Inequalities with Absolute Value

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3.5 Solving Equations and Inequalities with Absolute Value