# How do you solve x/(x-1)>2?

Jun 19, 2018

The solution is $x \in \left(2 , 4\right)$

#### Explanation:

You cannot do crossing over.

The inequality is

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

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

Placing on the same denominator

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

$\implies$, $\frac{x - 2 x + 4}{x - 2} > 0$

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

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

Let's build a 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}$$4$$\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 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}$$+$

$\textcolor{w h i t e}{a a a a}$$4 - x$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$color(white)(a)+$\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 a a}$$-$$\textcolor{w h i t e}{a a a a}$color(white)(a)+$\textcolor{w h i t e}{a a a a}$$-$

Therefore,

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

graph{(4-x)/(x-2) [-20.27, 20.27, -10.14, 10.14]}