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

Feb 19, 2017

The solution is x in ]-3,2[

#### Explanation:

We cannot do crossing over, so rewrite the inequality

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

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

Placing on the same denominator

$\frac{3 \left(x + 3\right) - 3 \left(x - 2\right)}{\left(x - 2\right) \left(x + 3\right)} \le 0$

$\frac{3 x + 9 - 3 x + 6}{\left(x - 2\right) \left(x + 3\right)} \le 0$

$\frac{15}{\left(x - 2\right) \left(x + 3\right)} \le 0$

Let $f \left(x\right) = \frac{15}{\left(x - 2\right) \left(x + 3\right)}$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{- 3 , 2\right\}$

Now, 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 a}$$- 3$$\textcolor{w h i t e}{a a a a a a}$$2$$\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}$$-$$\textcolor{w h i t e}{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}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 2$$\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}$$-$$\textcolor{w h i t e}{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}$$+$$\textcolor{w h i t e}{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}$$+$

Therefore,

$f \left(x\right) \le 0$ when x in ]-3,2[