How to solve the equality $|x−y|=|x|−|y|$ given that both $x$ and $y$ are of same sign and $|x|>|y|$

Question

I have tried using the method of $|x| = |x-y+y| < |x-y| + |y|$

so $|x| - |y| < |x - y|$

But the above statement not right coz the greater than sign comes in because of Triangle inequality

Answer 1

We have two cases

• $x,y>0 \quad |x|>|y| \implies x>y$

  $$|x−y|=x-y=|x|-|y|$$

• $x,y<0 \quad |x|>|y| \implies x<y$

  $$|x−y|=y-x=|x|-|y|$$

Answer 2

We can assume that $x\ge 0$ and $y\ge 0$.

then

$$|x|>|y|\implies x>y$$ $$ \implies |x-y|=x-y=|x|-|y|$$

it is always satisfied.