IF you really want to do algebra on these kinds of problems (which you shouldn't; as has been suggested above, you should always plug in numbers), you have to remember there are always 2 cases for an Inequality with variables in denominators:
x^2 + 1 / x > y? -->
1. Is x^2 + 1 > xy, while x is positive? or
2. Is x^2 + 1 < xy, while x is negative?
Then, your initial approach works: if x=y, then case 1 always works, but case 2 always doesn't because x^2+1 > x^2. However, Statement (2) invalidates the "what if x is negative?" possibility when combined with Statement (1), so the answer is C.
The only reason I mention this is that your initial logical approach was very good and it's always really satisfying on test day to see through the problem to the theory. If you can handle the weird logic of two possible superposed cases, then you CAN, in fact, do algebra with inequalities.
