If x is a real number, is x > 0? (1) |x| = -x (2) x + |x| > 0 Which of the following choic… — GMAT

If $x$ is a real number, is $x > 0$ ?

(1) $|x| = -x$
(2) $x + |x| > 0$

Which of the following choices 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.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.Cevap
E
Statements (1) and (2) TOGETHER are NOT sufficient.

EACH statement ALONE is sufficient.

The option stating that each statement alone is sufficient is correct. Statement (1) shows that

x \le 0

, which gives a definitive 'No' to whether

x > 0

. Statement (2) shows that

x > 0

, which gives a definitive 'Yes' to whether

x > 0

. Because both statements yield a clear, conclusive answer independently, each statement alone is sufficient.

Analyze Statement (1):

|x| = -x

By definition of absolute value,

|x| = -x

holds true if and only if

x \le 0

x \le 0

, then

x

is not strictly positive, so the answer to 'Is

x > 0

?' is a definitive 'No'.

Evaluate sufficiency of Statement (1)

Statement (1) provides a definitive 'No' answer to the question stem.

In Data Sufficiency, a statement that consistently yields a definitive 'No' is sufficient.

Analyze Statement (2):

x + |x| > 0

x \le 0

, then

|x| = -x

, so

x + |x| = x - x = 0

, which is not greater than

0

. If

x > 0

, then

|x| = x

, so

x + |x| = 2x > 0

The inequality

x + |x| > 0

is satisfied if and only if

x > 0

Evaluate sufficiency of Statement (2)

Statement (2) provides a definitive 'Yes' answer to the question stem.

Since Statement (2) guarantees

x > 0

, it is sufficient on its own.

Combine evaluations

Each statement alone provides a definitive answer to the question stem.

Therefore, EACH statement ALONE is sufficient.

Absolute Value Definition and Yes/No Data Sufficiency Decision Logic

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