Question

$\left[ \frac{x^{-1} - y^{-1}}{x^2 - y^2} \right]>1$?
I. $x + y>0$
II. $x$ and $y$ are positive integers and each is greater than 2.

• A. if the question can be answered with the help of statement I alone
• B. if the question can be answered with the help of statement II alone
• C. if both statement I and statement II are needed to answer the question
• D. if the statement cannot be answered even with the help of both the statements

Hint:

Always simplify algebraic expressions before checking sufficiency; sign analysis often directly gives the answer.

Answer 1

The Correct Option is B

We simplify the inequality: \[ \frac{x^{-1} - y^{-1}}{x^2 - y^2} = \frac{\frac{y - x}{xy}}{(x - y)(x + y)} = \frac{y - x}{xy(x - y)(x + y)}. \] Since \( y - x = -(x - y) \), this becomes: \[ \frac{-1}{xy(x + y)}. \] Thus, the expression is negative for positive \( x \) and \( y \). Therefore, it can never be \( > 1 \) if both \( x \) and \( y \) are positive integers greater than 2.

From Statement I: \( x + y > 0 \) is not enough because \( x \) and \( y \) could be negative or positive, so the sign and magnitude of the expression are uncertain. Hence, insufficient.

From Statement II: \( x, y \) are positive integers \( > 2 \), which makes the denominator positive and large, while the numerator is \(-1\), so the fraction is negative, meaning the inequality \( > 1 \) is false. Hence, sufficient to answer without I.