just as a double check, we want $y = \dfrac{x+4}{x-4} \geq 0$

There are 2 suspect points, where things go to 0, $x=-4, ~x=4$

So we split the real line up into $(-\infty, -4) \cup [-4] \cup (-4, 4) \cup [4] \cup (4, \infty)$

we can make a table of things

$\begin{matrix}

\text{Interval}&\text{Quotient}&\text{Result}\\

(-\infty, -4) &\dfrac N N &P\\

[-4] &\dfrac 0 N&0\\

(-4,4)&\dfrac P N &N\\

[4] &\dfrac P 0 &\text{not defined}\\

(4, \infty)&\dfrac P P &P

\end{matrix}$

and we see that the legit domain is

$(-\infty, -4] \cup (4, \infty)$

Which incidentally doesn't match your original post.