# How do you solve 6/(x+1)<-3?

May 24, 2017

The answer is $= x \in \left(- 3 , - 1\right)$

#### Explanation:

We cannot do crossing over.

Let's simplify the inequality

$\frac{6}{x + 1} < - 3$

$\frac{6}{x + 1} + 3 < 0$

$\frac{6 + 3 \left(x + 1\right)}{x + 1} < 0$

$\frac{6 + 3 x + 3}{x + 1} < 0$

$\frac{3 x + 9}{x + 1} < 0$

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

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

We can build 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}$$- 3$$\textcolor{w h i t e}{a a a a a a a}$$- 1$$\textcolor{w h i t e}{a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 3$$\textcolor{w h i t e}{a a a a a 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}$$+$

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

$\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}$$-$$\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 $x \in \left(- 3 , - 1\right)$