# How do you solve 4/(x-3)<2 using a sign chart?

Feb 15, 2017

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

#### Explanation:

Let's rearrange the equation

$\frac{4}{x - 3} - 2 < 0$

$\frac{4 - 2 \left(x - 3\right)}{x - 3} < 0$

$\frac{4 - 2 x + 6}{x - 3} < 0$

$\frac{10 - 2 x}{x - 3} < 0$

Let $f \left(x\right) = \frac{10 - 2 x}{x - 3}$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{3\right\}$

Now, we 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 a}$$3$$\textcolor{w h i t e}{a a a a a a a}$$5$$\textcolor{w h i t e}{a a a a a a}$$+ \infty$

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

$\textcolor{w h i t e}{a a a a}$$10 - 2 x$$\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 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}$$-$$\textcolor{w h i t e}{a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$

Therefore,

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