Answer
The domain is $(-\infty,-4]\cap [1,\infty)$.
Work Step by Step
To find the domain, we must solve the inequality $\dfrac{x-1}{x+4}\geq0$ since the values inside the radical can't be negative.
Now, we find critical points by equating to zero:
$x-1=0$
$x+4=0$
There are two critical points:
$x_1+4=0\rightarrow x_1=-4$
$x_2-1=0\rightarrow x_2=1$
Next, we are going to take three values: one less than -4; one between -4 and 1; and one more than 1 to test in the original equation and check if the inequality is true or not:
First test with a value less than -4:
$\dfrac{-5-1}{-5+4}\geq0$
$\dfrac{-6}{-1}\geq0$
$6\geq0 \rightarrow \text{ TRUE}$
Second test with a value between -4 and 1:
$\dfrac{-3-1}{-3+4}\geq0$
$\dfrac{-4}{1}\geq0$
$-4\geq0\rightarrow \text{ FALSE}$
Third test with a value more than 1:
$\dfrac{2-1}{2+4}\geq0$
$\dfrac{1}{6}\geq0 \rightarrow \text{ TRUE}$
These tests show that the domain of $\sqrt{\dfrac{x-1}{x+4}}$ is $(-\infty,-4]\cap [1,\infty)$
