# For what values of x is (x+2)/abs(x+2) <= 0 ?

Apr 8, 2017

$x \in \left(- 2 , - \infty\right)$

#### Explanation:

Note that for $\frac{x + 2}{\left\mid x + 2 \right\mid}$ to be defined $x \ne - 2$

Substitute $k = x + 2$

$\frac{x + 2}{\left\mid x + 2 \right\mid} = - 1$ (with the restriction that $x \ne - 2$)
$\Rightarrow k = - \left\mid k \right\mid$
This will be true for any non-negative value of $k$ (except for the case that would result in $x = - 2$)

That is $k \le 0$ (excluding any case where $x = - 2$)

$\Rightarrow x + 2 \le 0 \rightarrow x \le - 2$(excluding $x = - 2$)

$\Rightarrow x < - 2 \textcolor{w h i t e}{\text{XX}}$or equivalently$\textcolor{w h i t e}{\text{XX}} x \in \left(- 2 , - \infty\right)$