Okay, hope this works:
First, note that x+1 must be positive. Otherwise 1/(x+1) would be negative, and couldn't lie within the given range.
So,
1/2 > 1/(x+1)
x+1 > 2
x > 1
Note that we can multiply both sides of the inequality by x+1 without needing to worry about whether we need to reverse the inequality, because we know x+1 is positive. We also have:
1/5 < 1/x+1
x+ 1 < 5
x < 4
So 1 < x < 4, and x could be 2 or 3. You don't actually need to do all this work. If you look again at the original inequality: 1/5 < 1/(x+1) < 1/2, then x+1 must be between 2 and 5, which gives two possible integer values, 2 or 3.
From Statement 2, (x-3)(x-4)=0, we know that (x-3) times (x-4) is 0. If you ever multiply two things and get zero, you must have multiplied by zero. Thus x-3 = 0, or x-4 = 0, and x = 3 or 4. Note that this is the basis of the mathematical technique called 'factoring', when solving quadratic equations, and it's the reason we always set one side to be equal to zero before factoring a quadratic. It's a very important idea to understand for the GMAT. So, we again get two possible solutions.
Together, the only possible value that is consistent with both statements is x=3. So the answer should be C.
