I was solving the inequality $-4 \le \left|\frac {x+4} {2-x} \right| \le 4$ and I first wrote the domain, which is $(-\infty,-4] \cup (-4,2) \cup (2, \infty)$ and I got the solution that $x \le \frac 4 5$ and that $x \ge 4$, however the difference was my answer was $x \in (-4, \frac 4 5] \cup [4, \infty)$ but the correct answer is $x \in (-\infty, \frac 4 5] \cup [4, \infty)$.
Since the domain was defined as $(-4,2)$ for $x \le \frac 4 5$, how come the answer includes $-\infty$ which is outside it?
Your stated domain of $(-\infty,-4] \cup (-4,2) \cup (2, \infty)$ is not correct.
The only problematic value of $x$ in the middle expression is one that makes the denominator of the fraction $x-2$ equal to zero. In other words, the domain is $\Bbb R-\{2\}$, which in interval notation is
$$(-\infty,2) \cup (2,\infty)$$
The value of $-4$ is not a problem: it just makes the expression equal to zero, which is perfectly well defined--it fact, it even satisfies the inequality.
So the given correct answer $x \in (-\infty, \frac 4 5] \cup [4, \infty)$ is correct.