Answer to Question #4182 in Real Analysis for Junel

Question

If [i]a,b[/i]& are element of real numbers, show that [b]|a+b|=|a|+|b|[/b] & if [b]ab>=0[/b]

Answer 1

from the definition of |a+b|:|a+b|=a+b if a+b>=0 and |a+b|=-(a+b) if a+b<=0
from the condition ab>=0 we have two opportunities 1) a,b>=0 or 2) a,b<=0
so let's consider the case |a+b|=a+b if a+b>=0
a+b>=0, and 1) a,b>=0 or 2) a,b<=0 =>a,b>=0 Hence |a+b|=a+b=|a|+|b|
now the second case |a+b|=-(a+b) if a+b<0
a+b<=0 and 1) a,b>=0 or 2) a,b<=0 =>a,b<=0
Hence |a+b|=-(a+b)=-a-b=|a|+|b|
so we showed that |a+b|=|a|+|b| if ab >=0.