# How do you solve (x-4)/(x+3)<=0 using a sign chart?

Nov 12, 2016

The answer is x in ]-3,4]

#### Explanation:

As you canot divide by $0$, therefore $x \ne - 3$

Let $f \left(x\right) = \frac{x - 4}{x + 3}$

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}$$- 3$$\textcolor{w h i t e}{a a a a a}$$4$$\textcolor{w h i t e}{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}$$-$$\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 a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 4$$\textcolor{w h i t e}{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 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}$$+$$\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 a a}$$+$

Therefore, $f \left(x\right) \le 0$ $\implies$x in ]-3,4]
graph{(x-4)/(x+3) [-32.46, 32.46, -16.24, 16.25]}