The inequality boils down to
|y| > y
By definition of absolute value, we have
• |y| = y if y ≥ 0
• |y| = -y if y < 0
So if y ≥ 0, we have
y > y
but this is a contradiction.
On the other hand, if y < 0, we have
-y > y ==> 2y < 0 ==> y < 0
and no contradiction.
Now replace y with (x + 1)/(x - 1) + 1. Then you're left with solving
(x + 1)/(x - 1) + 1 < 0
(x + 1 + x - 1)/(x - 1) < 0
2x/(x - 1) < 0
The left side is negative if either 2x > 0 and x - 1 < 0, or 2x < 0 and x - 1 > 0. The first case reduces to x > 0 and x < 1, or 0 < x < 1. In the second case, we get x < 0 and x > 1, but x cannot satisfy both conditions, so we throw this case out.