How do you solve (x^2-4)/(3-x)>=0 using a sign chart?

Nov 19, 2016

The answer is x in] -oo,-2 ] uu [2, 3[

Explanation:

Let $f \left(x\right) = \frac{{x}^{2} - 4}{3 - x} = \frac{\left(x - 2\right) \left(x + 2\right)}{3 - x}$

The domain is ${D}_{f} = \mathbb{R} - \left\{3\right\}$

Let's do the sign chart

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

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

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

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

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

Therefore, $f \left(x\right) \ge 0$

when x in] -oo,-2 ] uu [2, 3[

graph{(x^2-4)/(3-x) [-34.51, 30.44, -18.83, 13.64]}