In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{x-6}{x+5}<0$

In the following exercises, we have to solve rational inequality and write the solution in interval notation.

$\frac{x-6}{x+5}<0$

Our inequality is in this form.

$\frac{x-6}{x+5}<0$

The rational expression will be zero when the numerator is zero. Since $x-6=0$when $x=6$, then $6$is a critical point.

The rational expression will be undefined when the denominator is zero. Since $x+5=0$when $x=-5$, then $-5$is a critical point.

The critical points are $6$and $-5$.

The number line is divided into three intervals:

$(-\infty ,-5)(-5,6)(6,\infty )$

To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.

$\mathit{I}\mathit{n}\mathit{t}\mathit{e}\mathit{r}\mathit{v}\mathit{a}\mathit{l}\mathbf{}\mathbf{}\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{5}\mathbf{)}$

The number $-6$is in the interval $(-\infty ,-5)$. Test $x=-6$in the expression in the numerator and the denominator.

$\frac{x-6}{x+5}=\frac{-6-6}{-6+5}=\frac{-12}{-1}=12$

The quotient positive in the interval $(-\infty ,-5)$.

The number $1$is in the interval $(-5,6)$. Test $x=1$in the expression in the numerator and the denominator.

$\frac{x-6}{x+5}=\frac{1-6}{1+5}=\frac{-5}{6}$

The quotient is marked negative in the interval $(-5,6)$.

The number $7$is in the interval $(6,\infty )$. Test in the expression in the numerator and the denominator.

$\frac{x-6}{x+5}=\frac{7-6}{7+5}=\frac{1}{12}$

The quotient is marked positive in the interval $(6,\infty )$.

We want the quotient to be less than zero, so the numbers in the intervals $(-5,6)$is the solution.

The solution in interval notation is $(-5,6)$