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