If x and y are positive real numbers, is \frac{1}{x} < \frac{1}{y}? (1) x^2 > y^2 (2) x - … — GMAT

If $x$ and $y$ are positive real numbers, is $\frac{1}{x} < \frac{1}{y}$ ?

(1) $x^2 > y^2$
(2) $x - y > 0$

Which of the following describes the sufficiency of the statements to answer the question?

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.Answer
E
Statements (1) and (2) TOGETHER are NOT sufficient.

EACH statement ALONE is sufficient.

Rephrasing the question stem

\frac{1}{x} < \frac{1}{y}

for positive numbers

x

and

y

yields

x > y

. Statement (1) states

x^2 > y^2

, which for positive numbers implies

x > y

(sufficient). Statement (2) states

x - y > 0

, which simplifies directly to

x > y

(sufficient). Therefore, each statement alone is sufficient.

Rephrase the question stem target inequality.

Since

x

and

y

are both positive, multiplying

\frac{1}{x} < \frac{1}{y}

xy

preserves the inequality sign, yielding

y < x

, which is equivalent to

x > y

. The target question simplifies to: 'Is

x > y

Simplifying the target question stem allows direct evaluation of each statement.

Evaluate Statement (1):

x^2 > y^2

Since

x

and

y

are positive real numbers, taking the positive square root of both sides of

x^2 > y^2

yields

x > y

. This gives a definitive 'Yes' to the rephrased question.

Statement (1) alone provides sufficient information.

Evaluate Statement (2):

x - y > 0

Adding

y

to both sides of

x - y > 0

gives

x > y

. This gives a definitive 'Yes' to the rephrased question.

Statement (2) alone provides sufficient information.

Question Stem Simplification and Target Rephrasing

Question