What's wrong with this method of trying to find the set of positive values of $x$ that satisfy the following inequality:
$\dfrac{1}{x}-\dfrac{1}{x-1} > \dfrac{1}{x-2}$
Find a common denominator on the left hand side:
$\dfrac{-1}{x(x-1)}>\dfrac{1}{x-2}\Leftrightarrow$
$2-x > x(x-1)\Leftrightarrow$
$2 > x^2$
Which is satisfied whenever $0 < x < \sqrt{2}$ (we only want positive $x$).
From our original inequality, we throw out the points $x=0, x=1, x=2$. But for $\sqrt{2} < x < 2$ we don't satisfy $x^2 < 2$, so I thought it should be $(0,1)\cup(1,\sqrt{2})$. This is wrong though... answer is $(0,1)\cup(\sqrt{2},2)$. Hope someone can help explain to me what steps were wrong. Maybe tell me explicitly which of the if and only if implications don't hold. Thanks a lot.