# How do solve (5x-8)/(x-5)>=2 and write the answer as a inequality and interval notation?

Jun 16, 2018

The solution is $x \in \left(- \infty , - \frac{2}{3}\right] \cup \left(5 , + \infty\right)$

#### Explanation:

The inequality is

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

We cannot do crossing over, so

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

Putting on the same denominator

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

$\frac{5 x - 8 - 2 x + 10}{x - 5} \ge 0$

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

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

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 a}$$- \frac{2}{3}$$\textcolor{w h i t e}{a a a a a a a}$$5$$\textcolor{w h i t e}{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}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 5$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$color(white)(aaaa)-$\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 a}$$+$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{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 $x \in \left(- \infty , - \frac{2}{3}\right] \cup \left(5 , + \infty\right)$ in interval notation

and $x \le - \frac{2}{3}$ and $x > 5$ as an inequality

graph{(5x-8)/(x-5)-2 [-29.8, 35.14, -12.44, 20.05]}