# How do you solve 3/(x+1)<=3 using a sign chart?

Jun 22, 2017

The solution is $x \in \left(- \infty , - 1\right) \cup \left[0 , + \infty\right)$

#### Explanation:

We cannot do crossing over

Let's rearrange the inequality

$\frac{3}{x + 1} \le 3$

$\frac{3}{x + 1} - 3 \le 0$

$\frac{3 - 3 x - 3}{x + 1} \le 0$

$\frac{- 3 x}{x + 1} \le 0$

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

Let's 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 a a}$$- 1$$\textcolor{w h i t e}{a 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}$$- 3 x$$\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}$$0$$\textcolor{w h i t e}{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}$$-$$\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}$$0$$\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}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a}$$-$

Therefore,

$f \left(x\right) \le 0$ when $x \in \left(- \infty , - 1\right) \cup \left[0 , + \infty\right)$ graph{3/(x+1)-3 [-19.3, 21.26, -12.45, 7.82]}