# How do you solve (3x+2)/(x-4)<0?

Jun 6, 2017

The solution is $x \in \left(- \frac{2}{3} , 4\right)$

#### Explanation:

Let $f \left(x\right) = \frac{3 x + 2}{x - 4}$

We build a 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}$$- \frac{2}{3}$$\textcolor{w h i t e}{a a a a a a a}$$4$$\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}$$3 x + 2$$\textcolor{w h i t e}{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}$$+$

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

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

Therefore,

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

graph{(3x+2)/(x-4) [-28.86, 28.85, -14.43, 14.45]}