Chapter 7.2, Problem 102E

To determine

To solve : the inequality $\left|x+10\right|\ge -3$

Answer to Problem 102E

The solution of inequality $\left|x+10\right|\ge -3$ is $-13\le x-\le 7$

Explanation of Solution

Given information :

Consider, the inequalityequation is,

  $\left|x+10\right|\ge -3$

Calculation :

The presence of absolute value means that $\left|f(x)\right|=\left|-f\left(x\right)\right|$ .

Therefore, in the inequality treat the absolute value as two different equation

When $x>0$ then $x+10\ge -3$ and

When $x<0$ then,

  $\begin{array}{l}-\left(x+10\right)\ge -3\\ -x-10\ge -3\end{array}$

The solution of first equation is

  $\begin{array}{l}x+10\ge -3\\ x+10-10\ge -3-10\left[\text{subtract 10 both side}\right]\\ x\ge -13\end{array}$

The solution of second equation is

  $\begin{array}{l}-x-10\ge -3\\ -x-10+x\ge -3+x\left[\text{add }x\text{ both side}\right]\\ -10\ge -3+x\\ -10+3\ge -3+x+3\left[\text{add 3 both side}\right]\\ -7\ge x\end{array}$

Hence, thesolution of inequality $\left|x+10\right|\ge -3$ is $-13\le x-\le 7$ .

The solution of the inequality in the real number line is shown below: