Solving an absolute value inequality with fractions I'm having a hard time figuring out this inequality: $\bigg|\dfrac{x-4}{x+5}\bigg| \le 4$
I use one of the absolute value properties, which results in: $-4 \leq \dfrac{x-4}{x+5} \leq 4$.
From there, I get $x \geq -\frac{16}{5}$ and $x \geq -8$, yet when I look at Wolfram alpha, the answer is $[-\frac{16}{5}, \infty)$ and $(-\infty, -8]$, which makes sense. What am I missing?
These are my steps to get $x \ge -8$:
\begin{align*}
&\frac{x - 4}{x + 5} \le 4 \\
&x - 4 \le 4(x + 5) \\
&x - 4 \le 4x + 20 \\
&x \le 4x + 24 \\
&-3x \le 24 \\
&x \ge -8.
\end{align*}

This is
$$y = \frac{x-4}{x+5} $$
$$ $$
The graph is a hyperbola, one horizontal asymptote and one vertical. I encourage you to get some graph paper and draw the same thing, maybe by plotting points when $x$  is an integer.  You can improve the picture by plotting points when $y$  is an integer, using
$$  x = \frac{5y+4}{-y+1}  $$
The educational bit: inequalities can be treacherous. A graph displays a good deal of  firm information, while the process of plotting one and drawing in the curve by hand cements some concepts that are otherwise a bit uncertain.
Your mapping is called a Mobius Transformation. There is a simple rule for finding the inverse function, it is another Mobius transformation.  Let me type in the rule, constants $a,b,c,d,$
$$  y = \frac{ax+b}{cx+d}  \Longrightarrow   x=\frac{dy-b}{-cy+a} $$
I figured out that Desmos will let me plot points. Here are enough points on the upper left arc of the hyperbola,  then just a few on the lower right. Next I'll put more points on the lower right...


