Question

Show that ||x|- |y|| < |x-y| , and please fully explain. Thnx

Answer 1

It should read that the difference is less than OR EQUAL TO

if both x and y are positive, these expressions are equal.

can prove it by looking at four possibilities: x positive or negative,
y positive or negative, and evaluating.

Answer 2

yeah the sign is actually </= , can you do it coz im clueless... Our lecturer ddnt explain anything... I dnt know where to start

Answer 3

What happens when x and y are both positive?

What happens when both are negative?

What happens when x is positive and y is negative?

What happens when x is negative and y is positive?

These are the four possibilities.

So, fore example, both positive: |x| = x, |y| = y so ||x|-|y|| = |x - y|

recall that |-x| = |x| and these both are positive

You must do the rest.

Answer 4

another trick is to prove that :
(|x|- |y|)^2 <= (x-y)^2

Answer 5

proving above is same as proving ur original statement

Answer 6

|x-y|^2 = (x-y)^2 and
=x ^2 - 2xy + y^2 and
=|x|^2 - 2|xy| + |y|^2
therfore: |x-y| >/= ||x| - |y||
.... Thats how i did it but im not sure if i obeyd the properties