How to solve $|x-5|=|2x+6|-1$? $|x-5|=|2x+6|-1$. 
The answer is $0$ or $-12$, but how would I solve it by algebraically solving it as opposed to sketching a graph?
$|x-5|=|2x+6|-1\\
(|x-5|)^2=(|2x+6|-1)^2\\
...\\
9x^4+204x^3+1188x^2+720x=0?$

That's not the way, because you're creating extra solutions (new roots) when you put square on both sides. Just separate the options:
$$x-5=2x+6-1\\
5-x=2x+6-1\\
x-5=-2x-6-1\\
5-x=-2x-6-1$$
Solve all of them, and you have the solutions
Take into acount that once you get your solutions, you have to check if they're possible, for example, in the first one, you have supposed that bot things inside || are positive, so if you get something for which x-5 or 2x+6 is negative, then you have to throw it away
