Expert: Paul Klarreich - 9/25/2009

Question
hi i'm really confused abt the triangle inequality..

i kinda understand the proof for |x + y| <= |x| + |y|, but for my homework answers which were provided, there is this part that goes

|(x + 2) − 4| ≤ (|x + 2| + 4)

i don't understand how this progression happened.. could you generalise your explanation/proof using x and y for me?

thx ^^

Answer
Questioner: momo
Country: Singapore
Category: Calculus
Private: No
Subject: triangle inequality
Question: hi i'm really confused abt the triangle inequality.

>> Please don't write 'IM-ese'.  This is not a tweet site.  Write the good English you learned in school.

i kinda understand the proof for |x + y| <= |x| + |y|, but for my homework answers which were provided, there is this part that goes

|(x + 2) − 4| ≤ (|x + 2| + 4)

i don't understand how this progression happened.. could you generalise your explanation/proof using x and y for me?

thx
...........................................
Let's write:

|(x + 2) − 4| ≤ (|x + 2| + 4)

as

|(A + 2) − 4| ≤ (|A + 2| + 4)

to avoid confusion with the x,y in |x + y| <= |x| + |y|, which is the T.I.

|(A + 2) − 4| = |(A + 2) + (− 4)|   << rewrite as a sum.

|(A + 2) + (− 4)| <= |A + 2| + |− 4|  << apply the T.I. with x=A+2, y=-4

|A + 2| + 4   <<  |-4| = 4.

How is that?