1. ## Basic inequality proof

Hi all,

I'm trying to do some self-studying to refresh and improve my maths skills but I'm struggling with an exercise to provide a proof in a specific way:
Noting that x=x+y-y, apply the theorem |x+y|<=|x|+|y| together with the fact that |-y|=|y| to prove that |x+y|>=|x|-|y|

I can see intuitively that it's true, but I haven't quite got the knack of manipulating magnitude inequalities. Any help would be much appreciated.

Thanks,

Chris

2. You have

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

You can write it like this

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

Now, you set x=u+v and y=-v

|u+v-v|-|-v| $\leq$ |u+v|

using that |-v|=|v|

|u| - |v| $\leq$ |u+v|

what is you want to prove.

3. $\left| x \right| = \left| {x - y + y} \right| \leqslant \left| {x - y} \right| + \left| y \right|$
Can you finish?

4. aha, so I can also just say |x| = |x + y + (-y)| <= |x + y| + |-y| and thus |x| - |y| <= |x + y|
Thanks a lot to you both!