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

Jan 15, 2017

The answer is x in ] -oo,0 [

#### Explanation:

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

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

$\forall x \in {D}_{f} \left(x\right) , {\left(x + 3\right)}^{2} > 0$

We can make 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 a a a}$$0$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\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}$$-$$\textcolor{w h i t e}{a a a a}$color(red)(∥)$\textcolor{w h i t e}{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 a}$color(red)(∥)$\textcolor{w h i t e}{a}$$+$

Therefore,

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