Chapter A, Problem 38E

To determine

To solve: The inequality.

Answer to Problem 38E

The interval notation for the given inequality is $\left(-\infty ,-4\right]\cup \left[2,\infty \right)$.

Explanation of Solution

Given:

The inequality is $\left|x+1\right|\ge 3$.

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,

$\begin{array}{l}\left|x+1\right|\ge 3\\ x+1\ge 3\text{ or }x+1\le -3\\ x\ge 3-1\text{ or }x\le -3-1\\ x\ge 2\text{ or }x\le -4\end{array}$

The corresponding interval notation is $\left(-\infty ,-4\right]\cup \left[2,\infty \right)$.

Thus, the interval notation for the given inequality is $\left(-\infty ,-4\right]\cup \left[2,\infty \right)$.