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The following is a snip from the book Olympiads: A Mathematical Olympiad Approach:

enter image description here

The obvious approach (for me) is to use the triangle inequality to get $$|a|-|b| \leq |a-b|$$ $$|b|-|a| \leq |b-a| = |a-b|$$

And the desired can be achieved by combining the two.

However, the given solution seems to do this much more succinctly. I have looked at this cryptic solution several times, but each time I've come back to it, I've never been able to make sense of it. Any help?

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Just one of those tricks that you need to convince yourself that are useful, then remember them for future problems you'll meet. – Asaf Karagila Feb 20 '12 at 0:25
I'm quite certain that what you wrote is the intended solution. – Bruno Joyal Feb 20 '12 at 0:26
up vote 2 down vote accepted

It's the same solution, just expressed slightly differently. $|a| = |a-b+b| \le |a-b| + |b|$ says $|a|-|b| \le |a-b|$. $|b|=|b-a+a|\le |b-a|+|a|$ says $|b|-|a|\le |b-a|$.

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