# How do you solve (5x-8)/(x-5)>=2 using a sign chart?

Feb 6, 2017

The answer is x in ]-oo, -2/3 ] uu ] 5, +oo[

#### Explanation:

Let's do some simplifications

$\frac{5 x - 8}{x - 5} \ge 2$

$\frac{5 x - 8}{x - 5} - 2 \ge 0$

$\frac{\left(5 x - 8\right) - 2 \left(x - 5\right)}{x - 5} \ge 0$

$\frac{3 x + 2}{x - 5} \ge 0$

Let $f \left(x\right) = \frac{3 x + 2}{x - 5}$

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}$$- \frac{2}{3}$$\textcolor{w h i t e}{a a a a}$$5$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$3 x + 2$$\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 a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 5$$\textcolor{w h i t e}{a 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 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 a}$$+$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

Therefore,

$f \left(x\right) \ge 0$ when x in ]-oo, -2/3 ] uu ] 5, +oo[