Question

Solve each rational function inequality and write the solution in interval notation.

Given the function$\mathrm{R}\left(\mathrm{x}\right)=\frac{(\mathrm{x}-5)}{(\mathrm{x}-2)}$find the values of x that make the function less than or equal to 0.

Short answer

Solution for inequality is$\mathrm{x}\in (2,5].$

Step-by-step solution

Step 1. Given Information

The given rational function is denoted as$\mathrm{R}\left(\mathrm{x}\right)=\frac{(\mathrm{x}-5)}{(\mathrm{x}-2)}.$

Step 2. Critical point

To find the critical points,

$$\begin{gathered} (\mathrm{x}-5)=0 \\ \mathrm{x}=5 \\ (\mathrm{x}-2)=0 \\ \mathrm{x}=2 \end{gathered}$$

To check the values,

$$\begin{gathered} \mathrm{At}\mathrm{x}=5,\mathrm{R}\left(\mathrm{x}\right)=\frac{0}{3},\mathrm{R}\left(\mathrm{x}\right)=0 \\ \mathrm{At}\mathrm{x}=2,\mathrm{R}\left(\mathrm{x}\right)=\frac{2-5}{0},\mathrm{R}\left(\mathrm{x}\right)=\infty \end{gathered}$$

When the polynomial function is undefined, the critical point will not add in interval notation.

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 number line sign of factors.

Thus,$\frac{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{5}\mathbf{)}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{)}}\mathbf{\le }\mathbf{0}\mathbf{}\mathbf{is}\mathbf{}\mathbf{true}\mathbf{}\mathbf{for}\mathbf{}\mathbf{x}\mathbf{\in }\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{5}\mathbf{]}\mathbf{.}$