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

You have $|x-5|=|2x+6|-1$. 
1)If $x\geq5$, $|x-5|=x-5$ and, $|2x+6|=2x+6$, so you have
$x-5=2x+6-1$
$x-5=2x+5$
$x=-10$ (but is not valid)
2)If $-3/2\leq x<5$, $|x-5|=-(x-5)$ and $|2x+6|=2x+6$, then
$-(x-5)=2x+6-1$
$-x+5=2x+5$
$-x=2x$
$x=0$
3)If $x<-3/2$, $|x-5|=-(x-5)$ and $|2x+6|=-(2x+6)$, then
$-(x-5)=-(2x+6)-1$
$-x+5=-2x-6-1$
$-x+5=-2x-7$
$-x+5=-2x-7$
$x=-12$
So, the answer for $x$ is $x=0$ or $x=-12$
