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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?

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Sometimes students concentrate too much on "algebra" (formal manipulation) and not enough on the geometry of the situation. The explanation by Chris Taylor, though phrased in terms of inequalities, probably comes from geometric insight: $|y-c|<\epsilon$ means that $y$ is at distance $<\epsilon$ from $c$. –  André Nicolas Jun 9 '11 at 16:01
    
Precisely that. –  Chris Taylor Jun 9 '11 at 16:18
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1 Answer

up vote 4 down vote accepted

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.

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My initial method was much more painstaking :P. I got the same answer (in twice as many steps) using $|x|=\sqrt{x^2}$. –  Pete Ley Jun 9 '11 at 16:06
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