Triangle Inequalities Relationship

1. Jun 6, 2012

ait.abd

I know the following
$$|x|-|y| \leq |x+y| \leq |x| + |y|$$
where does $$|x-y|$$ fit in the above equation?

2. Jun 6, 2012

algebrat

How about $x+(-y)$?

also, notice that there is a sort of better result, but I like the way you wrote it, makes it easier to remember and figure out what might be needed in a problem. But sometimes the left inequality is written:

$||x|-|y||\le|x+y|$

Just so that you understand when many other people write this.

3. Jun 6, 2012

ait.abd

So

$|x|-|y| \leq |x+y| \leq |x| + |y|$

and

$|x|-|y| \leq |x-y| \leq |x| + |y|.$

I think we can't say anything about the relationship between$|x+y|$ and $|x-y|,$
and in between $||x|-|y||$and $|x|-|y|.$

4. Jun 6, 2012

algebrat

$|x+y|\ge||x|-|y||\ge|x|-|y|$

5. Jun 6, 2012

theorem4.5.9

You can prove this pretty quickly by plugging numbers in, or just notice that the replacement $y \mapsto -y$ yields the other, hence the only possible relation is equality, which is clearly false.

6. Jun 16, 2012

vrmuth

if x and y have like signs then lx+yl ≥ lx-yl if unlike signs then lx+yl≤ lx-yl , check it out and always llxl-lyll >= lxl-lyl