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Really basic question here, but my maths is really rusty.

These things I know:

$$ a \ge | b - c | $$ $$ b \ge r $$ $$ c \le t $$

I'm trying to prove that given these things, the following is true:

$$ a \ge r - t $$

So I rearrange the things I know to get:

$$ a - |b - c| \ge 0 $$ $$ b - r \ge 0 $$ $$ t - c \ge 0 $$

Which I think means I can then add them together to get:

$$ a - |b - c| + b - c - r + t \ge 0 $$

If $b$ is always bigger than $c$, then $-|b-c|$ and $b-c$ cancel out which is then a simple rearrangement away from what we need to prove. However, $b$ isn't necessarily bigger than $c$, otherwise we wouldn't have bothered with the abs function.

So, what do I do to get those terms to go away?

P.S., I'm a programmer, is that still called abs in maths?

Mike Pierce
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sjmeverett
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1 Answers1

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Simply $$a\geq|b-c|\geq b-c\geq r-t$$

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    Feeling a bit stupid here, but I'm not completely sure I follow... – sjmeverett Aug 21 '13 at 21:32
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    The first unequality uses your first condition, the second uses the fact, that $|a|\geq a$ for all $a$ and the third one uses the second and third condition. – Tomas Aug 21 '13 at 21:39