Triangle Inequalities Relationship

[ait.abd]

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

 

[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.

 

[ait.abd]

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.

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|.

 

[algebrat]

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

 

[theorem4.5.9]

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

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.

 

[vrmuth]

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

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