- #1

r0bHadz

- 194

- 17

## Homework Statement

Prove |x|-|y| ≤ |x-y|

## Homework Equations

## The Attempt at a Solution

So you have 2 cases with 2 subcases in each

(1)

|x|-|y| ≤ x-y if x-y≥0

and

(2)

|x|-|y| ≤ -x+y if x-y≤0

(1.1) if x≥0 and y≥0, the result |x|-|y| = x-y is an obvious one

(1.2) if x≥0 and y≤0, |x|-|y| ≤ x-y because if y is not zero but less than zero, x-y will hold a greater value than |x|-|y|

(2.1) If x ≤0 y≥0, |x|-|y| = -x - y

(2.2) If x≤0 y≤0, |x|-|y| = -x - |-y| which ≤ -x+y

Are these all of the cases?

Not only that, is this proof valid? I feel like the method is pretty trivial. I don't see how it requires a "proof" seeing as you you have to know, for example, say a≤0, then |a| = -a, that's literally the only thing you need to know for this problem