triangle inequality

a)

Triangle inequality:

$|x|+|y|\ge |x+y|$.

Triangle inequality with $x=b$ and $y=a-b$:

$|b|+|a-b|\ge |b+a-b|=|a|$.

Rearranging yields

$|a-b|\ge |a|-|b|$,

as desired.

how did you just know that you had to do that??

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I rearranged the given inequality to get

$|a|\le |b|+|a-b|$ (1)

This resembles the triangle inequality, and from there we just equate the values in inequalities (1) and (2) (since they are in the same "triangle inequality" form).

We know that the triangle inequality is

$|x+y|\le |x|+|y|$, (2)

so we let

$\boxed{b=x}$

and

$\boxed{a-b=y}$,

then

$|x+y|=|b+(a-b)|=|a|$,

which satisfies (1), so we're done.