Show that |a+b| ≤ |a| + |b| and that |a| - |b| ≤ |a-b|

Question

The hint that my teacher gave me is to add the inequalities -|a| ≤ a ≤ |a| and -|b| ≤ b ≤ |b| as a system and to simplify it.

Answer 1

so, did you do it? If you add the inequalities, you get
-|a|-|b| ≤ a+b ≤ |a|+|b|
Now you just have to show that |a+b| ≤ a+b

Answer 2

how exactly would you show that? I can't find a way to simplify it any further for it to become |a+b| ≤ a+b

Answer 3

if a and b are positive, |a+b| = a+b, so that works
Let x = -a and y = -b, where a and b are positive
Then |x+y| = |-(a+b)| = -(-(a+b)) = a+b so that works
Now consider when a and b have opposite signs.