If x and y are real numbers such that x \neq 0, is |x| 0 Which of the following correctly describes the sufficiency of the statements?

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

If $x$ and $y$ are real numbers such that $x \neq 0$ , is $|x| < y$ ?

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

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

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.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Answer
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Combining both statements is sufficient. Statement (1) establishes that

|x| < |y|

because taking the square root of

x^2 < y^2

yields absolute values. Statement (2) specifies

y > 0

, meaning

|y| = y

. Substituting

y

for

|y|

gives

|x| < y

, which conclusively answers the question stem with 'Yes'.

Step-by-Step Solution

Rephrase the target question

The target question asks whether

|x| < y

. Since

|x| \ge 0

for all real numbers

x

, a necessary condition for

|x| < y

to be true is

y > 0

Simplifying absolute value conditions clarifies what values of variables are required.

Evaluate Statement (1):

x^2 < y^2

Taking the principal square root of both sides gives

\sqrt{x^2} < \sqrt{y^2}

, which simplifies to

|x| < |y|

The square root of a squared real number is its absolute value.

Test sufficiency of Statement (1) alone

x = 1

and

y = 2

, then

1^2 < 2^2

is true, and

|1| < 2

(Yes). If

x = 1

and

y = -2

, then

1^2 < (-2)^2

is true, but

|1| < -2

is false (No). Since both Yes and No are possible, Statement (1) alone is NOT sufficient.

Knowing

|x| < |y|

does not determine the sign of

y

Evaluate Statement (2):

y > 0

Statement (2) gives no information about

x

. For example, if

y = 5

and

x = 2

|2| < 5

(Yes). If

y = 5

and

x = 10

|10| < 5

is false (No). Thus Statement (2) alone is NOT sufficient.

No bound on

x

is provided.

Evaluate Statements (1) and (2) together

From Statement (1),

|x| < |y|

. From Statement (2),

y > 0

, which implies

|y| = y

. Substituting

|y| = y

into

|x| < |y|

gives

|x| < y

. This definitively answers 'Yes' to the question stem.

Combining the magnitude inequality with the sign constraint produces a unique, definitive Yes answer.

Key Concept

Absolute Value Inequalities and Number Ranges in Data Sufficiency

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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