I always liked doing the last question this way, not sure it's right though:
x3-y3 = (x-y)(x2+xy+y2)
When x > y,
x3-y3 is positive, x-y is positve, and as such x2+xy+y2 is positive to maintain the sign of the product, thus it is greater than 0.
When x < y,
x3-y3 is negative, x-y is...