# How do you solve (3x+5)/(6-2x)<=0?

Jan 19, 2017

The answer is x in ] -oo,-5/3 ] uu ] 3, +oo [

#### Explanation:

Let $f \left(x\right) = \frac{3 x + 5}{6 - 2 x}$

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

Now, we can make 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}$$- \frac{5}{3}$$\textcolor{w h i t e}{a a a a a a a}$$3$$\textcolor{w h i t e}{a a a a a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$3 x + 5$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a 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}$$6 - 2 x$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a 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 a a}$$-$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a}$∥$\textcolor{w h i t e}{a a a a}$$-$

Therefore,

$f \left(x\right) \le 0$, when x in ] -oo,-5/3 ] uu ] 3, +oo [