Show that if a and b have the same sign then:
|a + b| = |a| + |b|, and if a and b have opposite signs then:
|a + b| < |a| + |b|.
If you are talking about real numbers,
all you have to do is divide into cases...
(i) a>0, b<0, b<-a: |a|=a, |b|=-b, |a+b|=-(a+b)=-a-b<a-b=|a|+|b|.
(ii) a>0, b<0, b>=-a: |a|=a, |b|=-b, |a+b|=a+b<a-b=|a|+|b|.
(iii) a>0, b>0: ...