Answer
The domain is $(-\infty,-4]\cap [2,\infty)$
Work Step by Step
To find the domain, we must solve the inequality $\dfrac{x-2}{x+4}\geq0$ since the values inside the radical can't be negative.
Now, we find critical points by equating to zero:
$x-2=0$
$x+4=0$
There are two critical points:
$x_1+4=0\rightarrow x_1=-4$
$x_2-2=0\rightarrow x_2=2$
Next, we are going to take three values: one less than -4; one between -4 and 2; and one more than 2 to test in the original equation and check if the inequality is true or not:
First test with a value less than -4:
$\dfrac{-5-2}{-5+4}\geq0$
$\dfrac{-7}{-1}\geq0$
$7\geq0 \rightarrow \text{ TRUE}$
Second test with a value between -4 and 2:
$\dfrac{-2-2}{-2+4}\geq0$
$\dfrac{-4}{2}\geq0$
$-2\geq0\rightarrow \text{ FALSE}$
Third test with a value more than 2:
$\dfrac{3-2}{3+4}\geq0$
$\dfrac{1}{7}\geq0 \rightarrow \text{ TRUE}$
These tests show that the domain of $\sqrt{\dfrac{x-2}{x+4}}$ is $(-\infty,-4]\cap [2,\infty)$
