Proof of Triangle Inequality

[renolovexoxo]

Critique the following proof of the triangle inequality for real numbers x and y (|x+y|</=|x|+|y|)
When x and y are both non-negative, both sides equal x+y. When x and y are both non-positive, both sides equal -x-y. When x and y have opposite signs, we may assume that x>0>y. The inequality then holds because
|x+y|=max{x+y,-x-y}<x-y=|x|+|y|

I tried a critique twice and both times my teacher said I was wrong. Any help?

 

[guitarguy]

What do have so far for your proofs? And what can you learn from them?

Remember that |-x| = x and that |-x| + |-y| = x + y.

I want to help but I need more information describing what proof you are critiquing. For example, what are the steps of the proof to be critiqued? Keep trying! Trying twice is good I am sure you will get it.

 

[guitarguy]

When considering |x + y| I see a relationship that may help: the maximum occurs when x and y are of the same sign. The minumum occurs when x and y are of different signs. Try working this into your proof.

 

[renolovexoxo]

I have to critique the proof I stated, but I dont see what's there to critique.