# How do you solve x/(x-2)>9?

Feb 23, 2017

The solution is x in ]2,9/4[

#### Explanation:

We cannot do crossing over.

We rearrange the equation

$\frac{x}{x - 2} > 9$

$\frac{x}{x - 2} - 9 > 0$

$\frac{x - 9 \left(x - 2\right)}{x - 2} > 0$

$\frac{x - 9 x + 18}{x - 2} > 0$

$\frac{18 - 8 x}{x - 2} > 0$

$\frac{2 \left(9 - 4 x\right)}{x - 2} > 0$

Let $f \left(x\right) = \frac{2 \left(9 - 4 x\right)}{x - 2}$

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

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}$$- \infty$$\textcolor{w h i t e}{a a a a a a}$$2$$\textcolor{w h i t e}{a a a a a a a a}$$\frac{9}{4}$$\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 a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$9 - 4 x$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a 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 a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$

Therefore,

$f \left(x\right) > 0$ when x in ]2,9/4[