# How do you solve (x-2)/(x+2)<=2?

Aug 9, 2017

The solution is $x \in \left(- \infty , - 6\right] \cup \left(- 2 , + \infty\right)$

#### Explanation:

We cannot do crossing over, let's rewrite the inequality

$\frac{x - 2}{x + 2} \le 2$

$\frac{x - 2}{x + 2} - 2 \le 0$

$\frac{\left(x - 2\right) - 2 \left(x + 2\right)}{x + 2} \le 0$

$\frac{x - 2 - 2 x - 4}{x + 2} \le 0$

$\frac{- x - 6}{x + 2} \le 0$

$- \frac{x + 6}{x + 2} \le 0$

Let $f \left(x\right) = - \frac{x + 6}{x + 2}$

We can build the sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 6$$\textcolor{w h i t e}{a a a a a a a a}$$- 2$$\textcolor{w h i t e}{a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$- \left(x + 6\right)$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$\left(x + 2\right)$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a a a}$$-$$\textcolor{w h i t e}{a a}$$| |$$\textcolor{w h i t e}{a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a}$$-$

Therefore,

$f \left(x\right) \le 0$ when $x \in \left(- \infty , - 6\right] \cup \left(- 2 , + \infty\right)$

graph{(x-2)/(x+2)-2 [-22.8, 22.81, -11.4, 11.42]}