Section2.5Additional Practice Solving Absolute Value Equations and Inequalities

ExercisesExercises

Determine the solution set for each equation and inequality.

1.

$\abs{2x-4}=12$

Solution

The solution set is $\{-4,8\}\text{.}$

2.

$\abs{3-x}=-4$

Solution

The solution set is $\emptyset\text{.}$

3.

$\abs{5-7t}=11$

Solution

The solution set is $\left\{-\frac{6}{7},\frac{16}{7}\right\}\text{.}$

4.

$\abs{y+7}=7$

Solution

The solution set is $\{-14,0\}\text{.}$

5.

$\abs{\frac{3}{2}x-9}=9$

Solution

The solution set is $\{0,12\}\text{.}$

6.

$\abs{\frac{w+5}{3}}=15$

Solution

The solution set is $\{-50,40\}\text{.}$

7.

$\abs{2x+7} \le 14$

Solution

The solution set is $\left[-\frac{21}{2},\frac{7}{2}\right]\text{.}$

8.

$\abs{10-y} \lt 22$

Solution

The solution set is $(-12,32)\text{.}$

9.

$\abs{22+17x} \le -11$

Solution

The solution set is $\emptyset\text{.}$

10.

$\abs{\frac{4}{3}t+8} \le 24$

Solution

The solution set is $[-24,12]\text{.}$

11.

$\abs{\frac{8-x}{5}} \le 14$

Solution

The solution set is $[-62,78]\text{.}$

12.

$\abs{z+12} \le 0$

Solution

The solution set is $\{-12\}\text{.}$

13.

$\abs{3x+7} \ge -19$

Solution

The solution set is $(-\infty,\infty)\text{.}$

14.

$\abs{2x+31} \ge 17$

Solution

The solution set is $(-\infty,-24] \cup [-7,\infty)\text{.}$

15.

$\abs{-\frac{1}{2}w+7} \gt 21$

Solution

The solution set is $(-\infty,-28) \cup (56,\infty)\text{.}$

16.

$\abs{-5x+7} \ge 27$

Solution

The solution set is $(-\infty,-4] \cup \left[\frac{34}{5},\infty\right)\text{.}$

17.

$\abs{4x+2} \gt 0$

Solution

The solution set is $\left(-\infty,-\frac{1}{2}\right) \cup \left(-\frac{1}{2},\infty\right)\text{.}$

18.

$\abs{\frac{7-2x}{3}} \gt 4$

Solution

The solution set is $\left(-\infty,-\frac{5}{2}\right) \cup \left(\frac{19}{2},\infty\right)\text{.}$

19.

$5-\abs{2x+3}=-8$

Solution

The solution set is $\{-8,5\}\text{.}$

20.

$\abs{\frac{y+7}{3}}-8 \ge -4$

Solution

The solution set is $(-\infty,-19] \cup [5,\infty)\text{.}$

21.

$17-\abs{\frac{1-x}{3}} \ge 12$

Solution

The solution set is $[-14,16]\text{.}$

22.

$18+3\abs{-\frac{1}{2}t} \lt 24$

Solution

The solution set is $(-4,4)\text{.}$

23.

$12+\abs{w-9}=10$

Solution

The solution set is $\emptyset\text{.}$

24.

$\frac{-\abs{3+2x}+6}{2} \gt 1$

Solution

The solution set is $\left(-\frac{7}{2},\frac{1}{2}\right)\text{.}$