# How do you solve (2x+1) / (x-9) >= 0?

Mar 1, 2017

The solution is x in ]-oo, -1/2]uu]9,+oo[

#### Explanation:

We solve this inequality with a sign chart

Let $f \left(x\right) = \frac{2 x + 1}{x - 9}$

We can now 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}$$- \frac{1}{2}$$\textcolor{w h i t e}{a a a a a a}$$9$$\textcolor{w h i t e}{a a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$2 x + 1$$\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}$$x - 9$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a a a}$$-$$\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}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$

Therefore,

$f \left(x\right) \ge 0$ when x in ]-oo, -1/2]uu]9,+oo[