If x and y are real numbers such that x \neq 0, is \frac{|x - y|}{x} > 1? (1) x < 0 (2) y … — GMAT

If $x$ and $y$ are real numbers such that $x \neq 0$ , is $\frac{|x - y|}{x} > 1$ ?

(1) $x < 0$
(2) $y < 0$

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.

Statement (1) tells us

x < 0

. Since

|x - y| \ge 0

for all real numbers, dividing a non-negative numerator by a negative denominator guarantees that

\frac{|x - y|}{x} \le 0

. Consequently,

\frac{|x - y|}{x}

can never be greater than

1

, giving a definitive 'No' answer to the question. Because Statement (1) provides a single definitive answer, it is sufficient. Statement (2) tells us

y < 0

, but leaves the sign and value of

x

unconstrained. Choosing

x = -1

with

y = -2

gives a value of

-1

(answering 'No'), while choosing

x = 1

with

y = -2

gives a value of

3

(answering 'Yes'). Since Statement (2) allows both 'Yes' and 'No' outcomes, it is not sufficient. Thus, the option stating that Statement (1) alone is sufficient while Statement (2) alone is not sufficient is correct.

Step-by-Step Solution

Analyze the target question stem and algebraic constraints.

We are asked whether

\frac{|x - y|}{x} > 1

. Notice that for any real numbers

x

and

y

, the absolute value in the numerator

|x - y| \ge 0

. The sign of the denominator

x

determines the sign of the entire quotient.

Understanding the algebraic behavior of non-negative numerators over signed denominators simplifies statement evaluation.

Evaluate Statement (1):

x < 0

Since

|x - y| \ge 0

and

x < 0

, the ratio

\frac{|x - y|}{x}

is a non-negative number divided by a negative number. Thus,

\frac{|x - y|}{x} \le 0

. Since a number

\le 0

can never be greater than

1

, the answer to the question is a definitive 'No'. A definitive 'No' means the statement IS sufficient.

In Data Sufficiency, any statement that allows us to answer the question with a single, unambiguous 'Yes' or 'No' is sufficient.

Evaluate Statement (2):

y < 0

Test values for

x

while holding

y < 0

(e.g.,

y = -2

). If

x = -1

, then

\frac{|-1 - (-2)|}{-1} = \frac{1}{-1} = -1 \ngtr 1

(Answer: No). If

x = 1

, then

\frac{|1 - (-2)|}{1} = \frac{3}{1} = 3 > 1

(Answer: Yes). Because both 'Yes' and 'No' are possible, Statement (2) is not sufficient.

Getting conflicting answers from permissible test cases proves a statement is insufficient.

Key Concept

Definitive Yes/No logic and sign properties of absolute value quotients in Data Sufficiency

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Step-by-Step Solution

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