Question

Write the steps you would use to explain solving rational inequalities to your little brother.

Short answer

Solution is$\mathrm{x}\in (-\infty ,-4)\cup [-1,\infty ).$

Step-by-step solution

Step 1. Given data

Consider the rational inequality is$\mathrm{R}\left(\mathrm{x}\right)=\frac{(\mathrm{x}+1)}{(\mathrm{x}+4)}$.

Step 2. Critical points

Critical points are the points where rational expressions can be undefined or zero.

$$\begin{gathered} (\mathrm{x}+1)=0 \\ \mathrm{x}=-1 \\ (\mathrm{x}+4)=0 \\ \mathrm{x}=-4 \\ (\mathrm{x}+1)=0 \\ \mathrm{x}=-1 \\ (\mathrm{x}-4)=0 \\ \mathrm{x}=4 \end{gathered}$$

Step 3. Number line

On dividing the number line using the critical points,

Step 4. Intervals

There are three intervals includes $(-\infty ,-4),(-4,-1),(-1,\infty ).$

Take any value in the interval of $(-\infty ,-4).$

$$\begin{gathered} \mathrm{x}=-5 \\ \mathrm{R}\left(\mathrm{x}\right)=\frac{-5+1}{-5+4}=4 \\ \mathrm{R}\left(\mathrm{x}\right)=+\mathrm{ve} \end{gathered}$$

In the interval of $(-4,-1),\mathrm{take}\mathrm{x}=-2,$

$\mathrm{R}\left(\mathrm{x}\right)=\frac{-2+1}{-2+4}=\frac{-1}{2}=-\mathrm{ve}$

In the interval of$$\begin{gathered} (-1,\infty ),\mathrm{take}\mathrm{x}=5, \\ \mathrm{R}\left(\mathrm{x}\right)=\frac{5+1}{5+4}=\frac{2}{3}=+\mathrm{ve} \end{gathered}$$

Step 5. Number line

The number line can be drawn as,

Step 6. Testing of values

To test the values, the number line sign can indicated.

$$\begin{gathered} \mathrm{At}\mathrm{x}=-1,\mathrm{R}\left(\mathrm{x}\right)=\frac{-1+1}{-1+4}=0 \\ \mathrm{At}\mathrm{x}=-4,\mathrm{R}\left(\mathrm{x}\right)=\frac{-4+1}{-4+4}=\infty \\ \mathrm{Thus},\mathbf{R}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\ge }\mathbf{0}\mathbf{}\mathbf{x}\mathbf{\in }\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{4}\mathbf{)}\mathbf{\cup }\mathbf{[}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{\infty }\mathbf{)}. \end{gathered}$$