I'm trying to use some algebra get $||x-5|-10|<\epsilon$ into a more manageable form (I'd like it in terms of $0<|x+5|<\delta$) but I'm not sure where to begin. I don't really know the rules regarding absolute values within absolute values and can't seem to find anything that would help me on the net.
What are the rules pertaining to this sort of thing that I can use to get started messing around with this?
You can translate $|y-a| < b$ into $-b < y-a < b$, so you first get
$$-\epsilon < |x-5|-10 < \epsilon$$
which gives you the two inequalities
$$|x-5| > 10-\epsilon$$ $$|x-5| < 10 + \epsilon$$
and you can now apply similar rules to these two equations.