# How do you solve x/(x-2)>2 using a sign chart?

Dec 21, 2016

The answer is x in ] 2,4 [

#### Explanation:

We cannot do crossing over

$\frac{x}{x - 2} - 2 > 0$

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

$\frac{4 - x}{x - 2} > 0$

Let $f \left(x\right) = \frac{4 - x}{x - 2}$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{2\right\}$

Now we can do our sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$2$$\textcolor{w h i t e}{a a a a}$$4$$\textcolor{w h i t e}{a a a a}$$+ \infty$

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

$\textcolor{w h i t e}{a a a a}$$4 - x$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a}$∥$\textcolor{w h i t e}{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}$$-$$\textcolor{w h i t e}{a}$∥$\textcolor{w h i t e}{a}$$+$$\textcolor{w h i t e}{a a}$$-$

Therefore,

$f \left(x\right) > 0$ when x in ] 2,4 [