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

Look for where the expressions inside the absolute values change sign: $x-5$ changes sign at $x=5$, and $2x+6$ changes sign at $x=-3$. Thus, when $x<-3$, $x-5$ and $2x+6$ are both negative, and the equation is
$$-(x-5)=-(2x+6)-1\;.$$ 
When $-3\le x<5$, $x-5$ is negative and $2x+3$ is non-negative, so the equation is
$$-(x-5)=2x+6-1\;.$$
And when $x\ge 5$, both expressions are non-negative, and the equation is
$$x-5=2x+6-1\;.$$
Thus, you need to solve
$$\left\{\begin{align*}
&-x+5=-2x-7&&\text{when }x<-3\\
&-x+5=2x+5&&\text{when }-3\le x<5\\
&x-5=2x+5&&\text{when }x\ge 5\;.
\end{align*}\right.\tag{1}$$
$(1)$ reduces to
$$\left\{\begin{align*}
&x=-12&&\text{and }x<-3\\
&x=0&&\text{and }-3\le x<5\\
&x=-10&&\text{and }x\ge 5\;.
\end{align*}\right.\tag{2}$$
The first two solutions in $(2)$ fall within the intervals on which they are valid; the third does not and therefore is not a solution.
