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?
$|x-5|=|2x+6|-1\\
(|x-5|)^2=(|2x+6|-1)^2\\
...\\
9x^4+204x^3+1188x^2+720x=0?$

Consider different cases:
Case 1: $x>5$
In this case, both $x-5$ and $2x+6$ are positive, and you can resolve the absolute values positively. hence
$$
x-5=2x+6-1 \Rightarrow x = -10,
$$
which is not compatible with the assumption that $x>5$, hence no solution so far.
Case 2: $-3<x\leq5$
In this case, $x-5$ is negative, while $2x+6$ is still positive, so you get
$$
-(x-5)=2x+6-1\Rightarrow x=0;
$$
Since $0\in[-3,5]$, this is our first solution.
Case 3: $x\leq-3$
In this final case, the arguments of both absolute values are negative and the equation simplifies to
$$
-(x-5) = -(2x+6)-1 \Rightarrow x = -12,
$$
in agreement with your solution by inspection of the graph.
