# How do you solve (x-6)/(x+1)>0?

Apr 23, 2017

$\left(- \infty , - 1\right) \cup \left(6 , + \infty\right)$

#### Explanation:

$\text{the zero's of the numerator/denominator are}$

$\text{numerator"color(white)(x)x=6, "denominator} \textcolor{w h i t e}{x} x = - 1$

These indicate where the rational function may change sign and which value x cannot be on the denominator.

$\text{the intervals for consideration are}$

$x < - 1 , - 1 < x < 6 , x > 6$

$\text{Consider a "color(blue)"test point"" in each interval}$

$\text{we want to find where the function is positive, that is } > 0$

$\text{substitute each test point into the function and consider it's sign}$

$\textcolor{red}{x = - 2} \to \frac{-}{-} \to \textcolor{red}{\text{ positive}}$

$\textcolor{red}{x = 2} \to \frac{-}{+} \to \textcolor{b l u e}{\text{ negative}}$

$\textcolor{red}{x = 8} \to \frac{+}{+} \to \textcolor{red}{\text{ positive}}$

$\Rightarrow \left(- \infty , - 1\right) \cup \left(6 , + \infty\right)$
graph{(x-6)/(x+1) [-10, 10, -5, 5]}