# How do you solve ((3x-2)(x-4))/(x+4)^2<0 using a sign chart?

Jan 13, 2017

The answer is =x in ] 2/3 , 4 [

#### Explanation:

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

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{- 4\right\}$

The denominator is $> 0 , \forall x \in {D}_{f} \left(x\right)$

Let's do 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}$$\frac{2}{3}$$\textcolor{w h i t e}{a a a a a a}$$4$$\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}$$-$$\textcolor{w h i t e}{a a}$color(red)(∥)$\textcolor{w h i t e}{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}$$3 x - 2$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a}$color(red)(∥)$\textcolor{w h i t e}{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}$$+$$\textcolor{w h i t e}{a a}$color(red)(∥)$\textcolor{w h i t e}{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 ] 2/3 , 4 [