# How do you solve (2x)/(x-2)<=3 using a sign chart?

Jun 14, 2017

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

#### Explanation:

We cannot do crossing over

Let's rearrange the inequality

$\frac{2 x}{x - 2} \le 3$

$\frac{2 x}{x - 2} - 3 \le 0$

$\frac{2 x - 3 \left(x - 2\right)}{x - 2} \le 0$

$\frac{2 x - 3 x + 6}{x - 2} \le 0$

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

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

Let's 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}$$- \infty$$\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}$$6$$\textcolor{w h i t e}{a a 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}$$-$$\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}$$0$$\textcolor{w h i t e}{a a a a}$$+$

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

Therefore,

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