# How do you solve x/(x-2)>=0?

Mar 21, 2018

The solution is $x \in \left(- \infty , 0\right] \cup \left(2 , + \infty\right)$

#### Explanation:

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

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 a}$$0$$\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}$$x$$\textcolor{w h i t e}{a a a a 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}$$x - 2$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$color(white)(aaaaa)-$\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}$$+$$\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}$$| |$$\textcolor{w h i t e}{a a}$$+$

Therefore,

$f \left(x\right) \ge 0$ when 

graph{x/(x-2) [-10, 10, -5, 5]}

Mar 21, 2018

$\left(- \infty , 0\right]$ U $\left(2 , + \infty\right)$

#### Explanation:

x /(x - 2)≥0

x /(x - 2)≥0" : is true if" {("either", x ≥0 and x - 2 > 0),("or",x ≤ 0 and x - 2 < 0):}

x ≥0 and x - 2 > 0
$x > 2$

x ≤ 0 and x - 2 < 0
x ≤ 0

Answer: x ≤ 0 OR $x > 2$
In interval notation: $\left(- \infty , 0\right]$ U $\left(2 , + \infty\right)$