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

Dec 14, 2016

The answer is x in ] -oo,-2 [ uu ] 0,4]

#### Explanation:

The denominator is

${x}^{2} + 2 x = x \left(x + 2\right)$

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

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{0 , - 2\right\}$

Now, we can 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}$$- 2$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{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}$$x - 2$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a}$∥$\textcolor{w h i t e}{a}$$+$$\textcolor{w h i t e}{a a}$$+$$\textcolor{w h i t e}{a a a}$$+$

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

So,
$f \left(x\right) \le 0$, when x in ] -oo,-2 [ uu ] 0,4]