Question

Solve and write the solution in interval notation:

$\frac{x-2}{x+4}\ge 0$

Short answer

The solution in interval notation is$\left(-\infty ,-4\right)\cup [2,\infty )$

Step-by-step solution

Step 1. Given Information

Given inequality is $\frac{x-2}{x+4}\ge 0$

The inequality is to be solved and the solution is to be written in interval notation.

Step 2. Finding Critical Points

Critical points of a rational expression are the points where the expression becomes 0 or undefined.

The rational expression becomes 0 when the numerator is 0 and undefined when the denominator is 0.

When numerator is 0:

$$\begin{gathered} x-2=0 \\ x=2 \end{gathered}$$

When denominator is 0:

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

Hence the critical points are -4,2.

Step 3. Dividing the number line

Using the critical points, the number line is divided into intervals. The critical point of denominator is to be excluded as the boundary value in the interval and the critical value of the numerator is included in the interval.

Here the critical points are -4 and 2.

Hence the number line is divided into the intervals$\left(-\infty ,-4\right),(-4,2],[2,\infty )$

Step 4. Testing the intervals

One value from each interval is to be plugged into the inequality to check which interval satisfies the inequality.

For the interval $\left(-\infty ,-4\right)$,

Plugging$x=-6$:

$$\begin{gathered} \frac{x-2}{x+4}\ge 0 \\ \frac{-6-2}{-6+4}\stackrel{?}{\ge }0 \\ \frac{-8}{-2}\stackrel{?}{\ge }0 \\ 4\ge 0 \end{gathered}$$

Hence this interval is a solution to the inequality.

For the interval $(-4,2]$,

Plugging $x=0$:

role="math" localid="1645413875853" $$\begin{gathered} \frac{x-2}{x+4}\ge 0 \\ \frac{0-2}{0+4}\stackrel{?}{\ge }0 \\ \frac{-2}{4}\stackrel{?}{\ge }0 \\ -\frac{1}{2}\ngeqq 0 \end{gathered}$$

Hence this interval is not a solution to the inequality.

For the interval $[2,\infty )$,

Plugging $x=4$:

$$\begin{gathered} \frac{x-2}{x+4}\ge 0 \\ \frac{4-2}{4+4}\stackrel{?}{\ge }0 \\ \frac{2}{8}\stackrel{?}{\ge }0 \\ \frac{1}{4}\ge 0 \end{gathered}$$

Hence this interval is a solution to the inequality.

Step 5. Conclusion

The solution to the inequality in interval notation is$\left(-\infty ,-4\right)\cup [2,\infty )$