If x and y are non-zero real numbers, is \frac{x}{y} y Which of the following choices correctly describes the sufficiency of the statements?

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

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

(1) $|x + y| < |x - y|$
(2) $x > y$

Which of the following choices correctly describes the sufficiency of the statements?

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
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.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The correct option states that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Squaring both sides of Statement (1) gives

x^2 + 2xy + y^2 < x^2 - 2xy + y^2

, which simplifies directly to

4xy < 0

, meaning

xy < 0

. When two numbers have opposite signs, their quotient

\frac{x}{y}

must be negative, making it strictly less than 1. This provides a definitive 'Yes' answer. Statement (2) gives

x > y

; if

x = 3

and

y = 2

\frac{3}{2} > 1

(No), but if

x = 1

and

y = -2

\frac{1}{-2} = -0.5 < 1

(Yes). Hence Statement (2) is not sufficient.

Step-by-Step Solution

Rephrase the target question

The target question asks if

\frac{x}{y} < 1

Understanding the algebraic condition required for a fraction to be less than 1 helps evaluate given statements.

Evaluate Statement (1):

|x + y| < |x - y|

Squaring both sides yields

(x + y)^2 < (x - y)^2 \implies x^2 + 2xy + y^2 < x^2 - 2xy + y^2 \implies 4xy < 0 \implies xy < 0

Since both sides of the inequality are non-negative distance expressions, squaring preserves the inequality sign.

Determine the sufficiency of Statement (1)

Since

xy < 0

x

and

y

must have opposite signs. Therefore,

\frac{x}{y}

is negative, which means

\frac{x}{y} < 0 < 1

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

Any negative number is strictly less than 1, so Statement (1) alone is sufficient.

Evaluate Statement (2):

x > y

x = 3

and

y = 2

, then

x > y

and

\frac{x}{y} = 1.5 > 1

(No). If

x = 1

and

y = -2

, then

x > y

and

\frac{x}{y} = -0.5 < 1

(Yes).

Testing cases with positive vs. negative denominators shows that Statement (2) leads to both 'Yes' and 'No' answers.

Determine the sufficiency of Statement (2) and select the overall answer choice

Statement (2) alone is not sufficient. Therefore, Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

Only Statement (1) provides a definitive answer to the question stem.

Key Concept

Distance interpretation of absolute values and algebraic rephrasing of inequalities involving quotient sign analysis.

Estimated Time:2m 0s

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