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