show that $ \mid x - y\mid < \epsilon1 $ if and only if $x - \epsilon < y < x +\epsilon$

Question

show that $ \mid x - y\mid < \epsilon1 $ if and only if $x - \epsilon < y < x +\epsilon$

I know that $\mid x\mid < y $ => $-y < x < y$

I let $x = x -y$ and $y = \epsilon$

then I have $-\epsilon < \mid x - y \mid < +\epsilon $

I tried to use a triangle equlity, but $\mid x- y \mid \le \mid x \mid + \mid y \mid $

but it doenst make sense...

I dont know how to solve the problem from my last step.

Answer 1

We have that

$$|x-y|=|y-x|.$$

Then

$$|x-y|<\epsilon \iff -\epsilon < y - x < \epsilon\iff x-\epsilon < y < x+\epsilon.$$

In the last step, we merely added $x$ to each inequality.