Chapter A, Problem 37E

To determine

To solve: The inequality.

Answer to Problem 37E

The interval notation for the given inequality is $\left(-\infty ,-7\right]\cup \left[-3,\infty \right)$.

Explanation of Solution

Given:

The inequality is $\left|x+5\right|\ge 2$.

Property used:

$\left|x\right|<a$ if and only if $-a<x<a$.

Calculation:

By the property mentioned above, the given function can be re written as,

$\begin{array}{l}\left|x+5\right|\ge 2\\ x+5\ge 2\text{ or }x+5\le -2\\ x\ge 2-5\text{ or }x\le -2-5\\ x\ge -3\text{ or }x\le -7\end{array}$

The corresponding interval notation is $\left(-\infty ,-7\right]\cup \left[-3,\infty \right)$.

Thus, the interval notation for the given inequality is $\left(-\infty ,-7\right]\cup \left[-3,\infty \right)$.