# How do you solve and write the following in interval notation: (4-x) /( x-8)>=0?

Jan 21, 2017

The answer is x in [ 4, -8 [

#### Explanation:

Let $f \left(x\right) = \frac{4 - x}{x - 8}$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{8\right\}$

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}$$- \infty$$\textcolor{w h i t e}{a a a a}$$4$$\textcolor{w h i t e}{a a a a a a a a}$$8$$\textcolor{w h i t e}{a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$4 - x$$\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}$$\textcolor{red}{| |}$$\textcolor{w h i t e}{a a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$x - 8$$\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}$$\textcolor{red}{| |}$$\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}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$\textcolor{red}{| |}$$\textcolor{w h i t e}{a a a a}$$-$

Therefore,

$f \left(x \ge 0\right)$ when x in [ 4, -8 [