Question

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

$\frac{1}{{\mathrm{x}}^2-16}<0$

Short answer

Solution for rational inequality is$\mathrm{x}\in (-4,4).$

Step-by-step solution

Step 1. Given Information

Given that the rational inequality is$\frac{1}{{\mathrm{x}}^2-16}<0.$

Step 2. Algebraic Expression simplification

Take $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathrm{x}}^2-16}$

On applying formula and simplifying algebraic function,

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\frac{1}{({\mathrm{x}}^2-4^2)} \\ ({\mathrm{x}}^2-4^2)=(\mathrm{x}+4)(\mathrm{x}-4)\because ({\mathrm{a}}^2-{\mathrm{b}}^2)=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b}) \\ \mathrm{f}\left(\mathrm{x}\right)=\frac{1}{(\mathrm{x}-4)(\mathrm{x}+4)} \end{gathered}$$

Step 3. Critical points

To find the critical points,

$$\begin{gathered} (\mathrm{x}-4)=0 \\ \mathrm{x}=4 \\ (\mathrm{x}+4)=0 \\ \mathrm{x}=-4 \end{gathered}$$

There is an undefined polynomial function using this critical points. Thus, it will not be taken.

Divide the number line and test the polynomial expression sign in intervals.

In $$\begin{gathered} \frac{1}{{\mathrm{x}}^2-16}<0\mathrm{in}\mathrm{interval}(-4,4) \\ \mathrm{f}\left(\mathrm{x}\right)>0,\mathbf{x}\mathbf{\in }\mathbf{(}\mathbf{-}\mathbf{4}\mathbf{,}\mathbf{4}\mathbf{)} \end{gathered}$$