91Ó°ÊÓ

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

$\frac{x+5}{x−2}<0$

Our inequality is in this form.

$\frac{x+5}{x−2}<0$

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

The rational expression will be undefined when the denominator is zero. Since $x-2=0$when $x=2$, then $2$is a critical point.

The critical points are $-5$and $2$.

The number line is divided into three intervals:

$(-∞,-5)(-5,2)(2,∞)$

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.

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

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

$\frac{x+5}{x−2}=\frac{-6+5}{-6−2}=\frac{-1}{-8}=\frac{1}{8}$

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

The number $1$is in the interval $(-5,2)$. Test $x=1$in the expression in the numerator and the denominator.

$\frac{x+5}{x−2}=\frac{1+5}{1−2}=\frac{6}{-1}=-6$

The quotient is marked negative in the interval $(-5,2)$.

The number $3$is in the interval $(2,∞)$. Test $x=3$in the expression in the numerator and the denominator.

$\frac{x+5}{x−2}=\frac{3+5}{3−2}=\frac{8}{1}=8$

The quotient is marked negative in the interval $(2,∞)$.

We want the quotient to be greater than or equal to zero, so the numbers in the intervals $(-∞,-5)$and $(2,∞)$are solutions.

The solution in interval notation is$(-∞,-5)∪(2,∞)$