If x and y are real numbers, is |x - y| 0 (2) |x| < y

If x and y are real numbers, is |x - y| < |x + y|? (1) x^3 y > 0 (2) |x| < y — GMAT

If $x$ and $y$ are real numbers, is $|x - y| < |x + y|$ ?

(1) $x^3 y > 0$
(2) $|x| < y$

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.

Rephrasing the stem by squaring both non-negative sides reveals that

|x - y| < |x + y|

is strictly equivalent to

xy > 0

. Statement (1) specifies

x^3 y > 0

. Because

x^3

and

x

always have the same sign,

x^3 y > 0

means

x

and

y

share the same sign, which guarantees

xy > 0

. Thus, Statement (1) alone provides a definitive 'Yes'. Statement (2) states

|x| < y

, which forces

y > 0

but allows

x

to be either positive or negative, so

xy

can be either positive or negative. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

Rephrase the question stem by squaring both sides of the inequality.

Since both

|x - y|

and

|x + y|

are non-negative,

|x - y| < |x + y| \iff (x - y)^2 < (x + y)^2 \iff x^2 - 2xy + y^2 < x^2 + 2xy + y^2 \iff 0 < 4xy \iff xy > 0

. The question asks whether

x

and

y

have the same sign (and neither is zero).

Simplifying absolute value inequalities before analyzing statements prevents misinterpretation.

Evaluate Statement (1):

x^3 y > 0

Since

x^3

always has the exact same algebraic sign as

x

for all real numbers,

x^3 y > 0 \implies xy > 0

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

Statement (1) alone provides enough information to answer the target question conclusively.

Evaluate Statement (2):

|x| < y

Since

|x| \ge 0

, this implies

y > 0

. However,

x

can be positive (e.g.,

x=2, y=3 \implies xy=6 > 0

, Yes) or negative (e.g.,

x=-2, y=3 \implies xy=-6 < 0

, No).

Because Statement (2) allows both 'Yes' and 'No' answers, it is not sufficient.

Key Concept

Rephrasing absolute value inequalities and analyzing product sign constraints in Data Sufficiency

Question

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Step-by-Step Solution

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