Help with showing that $|x+y|=|x|+|y| \longleftrightarrow xy\ge0 $

Question

I need to prove that $|x+y|=|x|+|y| \longleftrightarrow xy\ge0 $.

I proved before that for any $x,y\in R$ holds $|x+y|\le|x+y|$ and I thought maybe it could help me with my arguments to show what I need. On the other hand if $xy\ge0$ then it means that both $x,y$ are positive or negative. I don't see exactly how it helps me somehow.

Any hints on how should I proceed?

score 8 · Accepted Answer · answered Sep 09 '13 at 11:05

8

Square (note that both sides are non-negative!):

$$|x+y|=|x|+|y|\iff x^2+2xy+y^2=x^2+2|xy|+y^2\iff $$

$$xy=|xy|\stackrel{\text{by def.}}\iff xy\ge 0$$

answered Sep 09 '13 at 11:05

DonAntonio

211,718
17
136
287

2

Short and sweet! +1 – amWhy Sep 09 '13 at 11:05
not sure what you did here? do you square here both sides? – Yossi Sep 09 '13 at 11:21
certainly you cannot square just one side :-) Remember that $|a|^2 = a^2$ for each $a$. – mau Sep 09 '13 at 11:49
Yes, I did square both sides. As opposed to many other cases, in this one we don't have the possible problem about signs since, as mentioned, both sides are positive... – DonAntonio Sep 09 '13 at 12:14

Help with showing that $|x+y|=|x|+|y| \longleftrightarrow xy\ge0 $

1 Answers1