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

##### 1 Answer
Jan 18, 2017

The answer is x in [-1, 0 [

#### Explanation:

We cannot cross over.

Let's rewrite the inequality

$\frac{x + 3}{x} + 2 \le 0$

$\frac{x + 3 + 2 x}{x} \le 0$

$\frac{3 x + 3}{x} \le 0$

$\frac{3 \left(x + 1\right)}{x} \le 0$

Let $f \left(x\right) = \frac{3 \left(x + 1\right)}{x}$

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}$$- 1$$\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 a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 1$$\textcolor{w h i t e}{a a a a}$$-$$\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}{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}$$-$$\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}{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 a a a}$$-$$\textcolor{w h i t e}{a a}$∥$\textcolor{w h i t e}{a a}$$+$

Therefore,

$f \left(x\right) \le 0$, when x in [-1, 0 [