Question

Solve each rational function inequality and write the solution in interval notation. Given the function$\mathrm{R}\left(\mathrm{x}\right)=\frac{(\mathrm{x}-6)}{(\mathrm{x}+2)}$, find the values of x that make the function less than or equal to 0.

Short answer

Solution for the rational function is$\mathrm{x}\in (-2,6].$

Step-by-step solution

Step 1. Given Data

Given that the rational function is$\mathrm{R}\left(\mathrm{x}\right)=\frac{(\mathrm{x}-6)}{(\mathrm{x}+2)}.$

Step 2. Critical Point

To find the critical point,

$$\begin{gathered} (\mathrm{x}-6)=0 \\ \mathrm{x}=6 \\ (\mathrm{x}+2)=0 \\ \mathrm{x}=-2 \\ \mathrm{At}\mathrm{x}=6,\mathrm{R}\left(\mathrm{x}\right)=\frac{6-6}{6+2}=0 \\ \mathrm{At}\mathrm{x}=-2,\mathrm{R}\left(\mathrm{x}\right)=\frac{-2-6}{-2+2}=\infty \end{gathered}$$

There is an undefined polynomial function of critical point will not be taken.

Step 3. Number line

On dividing the number line using critical point,

Step 4. Testing of polynomial expression

On testing the sign of polynomial expression and it has indicated.

$\frac{(\mathrm{x}-6)}{(\mathrm{x}+2)}\le 0;(-2,6]$

Thus,$\mathbf{R}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\le }\mathbf{0}\mathbf{}\mathbf{for}\mathbf{}\mathbf{x}\mathbf{\in }\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{6}\mathbf{]}\mathbf{.}$