Question

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

$\frac{4}{{\mathrm{x}}^2-25}>0$

Short answer

Solution for rational inequality is$\mathrm{x}\in (-\infty ,-5)\cup (5,\infty ).$

Step-by-step solution

Step 1. Given Information

Given that the rational inequality is$\frac{4}{{\mathrm{x}}^2-25}>0.$

Step 2. Polynomial factor

On factoring the polynomial,

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\frac{4}{{\mathrm{x}}^2-25} \\ \mathrm{f}\left(\mathrm{x}\right)=\frac{4}{{\mathrm{x}}^2-5^2} \\ \mathrm{f}\left(\mathrm{x}\right)=\frac{4}{(\mathrm{x}-5)(\mathrm{x}+5)} \end{gathered}$$

Step 3. Critical point

To find the critical points,

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

There is an undefined polynomial function at this critical points. It will not be included. On dividing the number line,

To test the polynomial expression sign, it indicates the number line sign.

$$\begin{gathered} \frac{1}{{\mathrm{x}}^2-16}<0,(-\infty ,-5)\cup (5,\infty ) \\ \mathrm{f}\left(\mathrm{x}\right)<0,\mathbf{x}\mathbf{\in }\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{5}\mathbf{)}\mathbf{\cup }\mathbf{(}\mathbf{5}\mathbf{,}\mathbf{\infty }\mathbf{)} \end{gathered}$$