When does the triangle inequality hold for absolute value?

[lepton123]

Homework Statement

abs(x+y+z)≤abs(x)+abs(y)+abs(z) indicate when this equality holds and prove this statement

Homework Equations

Triangle inequality?

The Attempt at a Solution

I have nothing :/

 

[UltrafastPED]

What is the absolute value of x? if x => 0 then abs(x)=x, else it is -x.

So make a table with all of the possible cases and see what happens!

 

[lepton123]

Drat, I was hoping that that I wouldn't have to do a case analysis; is there a more elegant way of solving this though?

 

[UltrafastPED]

I don't know elegant ... I grew up on a farm!

So once you have carried out the detailed work you can apply your own standards of elegance and cleverness ... and write something elegant!

 

[verty]

Assume (by renaming of variables) that x ≤ y ≤ z. Then you have 4 cases to check, it shouldn't be too tedious.

 

[JanEnClaesen]

x + y + z <= |x| + |y| + |z|
- x - y - z <= |x| + |y| + |z|

(definition of the absolute value)