Is ( |x-1y-1| )-1 > xy?
(1) xy > 1
(2) x2 > y2
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
We can rephrase the question by manipulating it algebraically:
(|x-1 * y-1|)-1 > xy
(|1/x * 1/y|)-1 > xy
(|1/xy|)-1 > xy
1/(|1/(xy)|) > xy
Is |xy| > xy?
The question can be rephrased as “Is the absolute value of xy greater than xy?” And since |xy| is always positive, this is only true when xy < 0. If xy > 0, |xy| = xy. Therefore, this question is really asking whether xy < 0, i.e. whether x and y have opposite signs.
(1) SUFFICIENT: If xy > 1, xy is definitely positive. For xy to be positive, x and y must have the same sign, i.e. they are both positive or both negative. Therefore x and y definitely do not have opposite signs and |xy| is equal to xy, not greater. This is an absolute "no" to the question and therefore sufficient.
(2) INSUFFICIENT: x2 > y2
Algebraically, this inequality reduces to |x| > |y|. This tells us nothing about the sign of x and y. They could have the same signs or opposite signs.
The correct answer is A: Statement (1) alone is sufficient, but statement (2) alone is not.