# How do you solve the inequality  |x + 2| ≤ 3 ?

Feb 8, 2016

$x \in \left[- 5 , 1\right]$

#### Explanation:

$| x + 2 | \le 3$

We must divide the equation in two parts:

If $x + 2 \ge 0$:

The module of a positive number is the same number, the equation gives:

$x + 2 \le 3$ and $x + 2 \ge 0$:
$x \le 1$ and $x \ge - 2$:

$x \in \left[- 2 , 1\right]$

If $x + 2 \le 0$:

The module of a negative number is its inverse , the equation gives:

$- \left(x + 2\right) \le 3$ and $x + 2 \le 0$:

$- x - 2 \le 3$ and $x \le - 2$:

$- x \le 5$ and $x \le - 2$:

Now we must multiply x by -1. This will change the comparision sign.

$x \ge - 5$ and $x \le - 2$,

$x \in \left[- 5 , - 2\right]$

The solution will be the union of the two intervals:

$x \in \left[- 5 , 1\right]$