91Ó°ÊÓ

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

$\frac{4x−1}{x−4}≤1$

The given function is$\frac{4x−1}{x−4}≤1$

Subtract by $1$on both side

$$\begin{gathered} \frac{4x−1}{x−4}-1≤1-1 \\ \frac{4x−1}{x−4}-1≤0 \end{gathered}$$

Simplifying the function

$$\begin{gathered} \frac{4x−1}{x−4}-\frac{1(x−4)}{x−4}≤0 \\ \frac{4x−1-1(x−4)}{x−4}≤0 \\ \frac{4x−1-x+4}{x−4}≤0 \\ \frac{3x+3}{x−4}≤0 \end{gathered}$$

The rational expression will be zero when the numerator is zero. Since $3x+3=0$when $x=-1$, then $-1$is a critical point.

The rational expression will be undefined when the denominator is zero. Since $x-4=0$when $x=4$, then $4$is a critical point.

The critical points are $-1$and $4$.

The number line is divided into three intervals:

$(-∞,-1)(-1,4)(4,∞)$

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.

$\mathit{I}\mathit{n}\mathit{t}\mathit{e}\mathit{r}\mathit{v}\mathit{a}\mathit{l}\mathbf{}\mathbf{}\mathbf{}\mathbf{(}\mathbf{-}\mathbf{∞}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{)}$

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

$\frac{3x+3}{x−4}=\frac{3×(-2)+3}{(-2)−4}=\frac{-6+3}{-6}=\frac{-3}{-6}=\frac{1}{2}$

The number $1$is in the interval $(-1,4)$. Test $x=1$in the expression in the numerator and the denominator.

$\frac{3x+3}{x−4}=\frac{3×1+3}{1−4}=\frac{3+3}{−3}=\frac{6}{-3}=-2$

The quotient is marked negative in the interval $(-1,4)$.

The number $5$is in the interval $(4,∞)$. Test $x=5$in the expression in the numerator and the denominator.

$\frac{3x+3}{x−4}=\frac{3×5+3}{5−4}=\frac{15+3}{1}=18$

The quotient is marked positive in the interval $(4,∞)$.

We want the quotient to be less than or equal to zero, so the numbers in the interval$(-1,4)$is the solution.

The solution in interval notation is $(-1,4)$