Chapter 3.3, Problem 39E

To determine

To find: the values of x that satisfies the inequality.

Answer to Problem 39E

  $-17\le x\le 7$

Explanation of Solution

Given information:

  $-\frac{2}{3}\left|x+5\right|\ge -8$

Calculation:

The outcomes of the given equality can be either of the two shown below.

  $-\frac{2}{3}\left(x+5\right)\ge -8$..........(1)

  $-\frac{2}{3}\left[-\left(x+5\right)\right]\ge -8$..........(2)

Solve inequality (1) as follows:

  $\begin{array}{l}-\frac{2}{3}\left(x+5\right)\ge -8\\ -\frac{2x}{3}-\frac{10}{3}\ge -8\\ -\frac{2x}{3}-\frac{10}{3}\ge -\frac{8}{3}\left(3\right)\left[\text{Taking common denominator}\right]\\ -2x-10\ge -24\\ -2x-10+10\ge -24+10\left[\text{Add 10 on both the sides}\right]\\ -2x\ge -14\\ \therefore x\le 7\left[\text{Dividing both sides by }-\text{2}\right]\end{array}$

Solve inequality (2) as follows:

  $\begin{array}{l}-\frac{2}{3}\left[-\left(x+5\right)\right]\ge -8\\ -\frac{2}{3}\left[-x-5\right]\ge -8\\ -\frac{2\left(3\right)}{3}\left[-x-5\right]\ge -8\left(3\right)\left[\text{Multiply both sides by 3}\right]\\ -6\left(-x-5\right)\ge -24\\ 6x+30\ge -24\\ 6x+30-30\ge -24-30\left[\text{Subtract both sides by }30\right]\\ 6x\ge -54\\ 6x\ge -54\left[\text{Divide both sides by 6}\right]\\ \therefore x\ge -17\end{array}$

Thus, the solution of given inequality is $-17\le x\le 7$ .