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.Cevap
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.

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

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