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Section 1.5 Additional Practice Solving Absolute Value Equations and Inequalities

Exercises Exercises

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 \(\left(-\frac{7}{2},\frac{1}{2}\right)\text{.}\)