How to solve $|1+1/x| > 2$ I am having a problem with this question:

Find all real $x$ that satisfies $|1 + 1/x| > 2$.

This is clearly not defined in $x = 0$.
By my logic, it should be solved with:

$1+1/x > 2$   or  $1+1/x < -2$

But the result I am getting from this is wrong. ($x<1$)
Correct result is 

$-1/3 < x < 1$

How to solve a problem like this? Why is this logic not working here:

$|x| > 2$
$x > 2$ or $x < -2$


Alternatively, $x\ne 0$ and:
$$|x+1|>2|x| \iff (x+1)^2>(2x)^2 \iff (x-1)(3x+1)<0 \iff -1/3<x<1$$
Hence:
$$x\in (-1/3,0)\cup (0,1)$$
