If x is a real number, is x > 0? (1) x^3 - x = 0 and x \neq 0 (2) |x + 2| = -x - 2 Which o… — GMAT

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

(1) $x^3 - x = 0$ and $x \neq 0$
(2) $|x + 2| = -x - 2$

Which of the following choices correctly identifies the sufficiency of the statements?

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Answer
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.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (2) alone leads to

x \le -2

, which guarantees that

x

cannot be positive. Hence, the question 'Is

x > 0

?' receives a definitive 'No', making Statement (2) alone sufficient. Statement (1) allows

x

to be either

1

(Yes) or

-1

(No), making it insufficient.

Analyze the target question type

The target is a Yes/No Data Sufficiency question: 'Is

x > 0

?' A statement is sufficient if it yields a definitive 'Yes' (always true) OR a definitive 'No' (always false).

Understanding Yes/No Data Sufficiency structure prevents misinterpreting a definitive 'No' as insufficient.

Evaluate Statement (1) independently

x^3 - x = 0 \implies x(x^2 - 1) = 0 \implies x(x - 1)(x + 1) = 0

. Given

x \neq 0

x = 1

x = -1

.
If

x = 1

, then

x > 0

is YES.
If

x = -1

, then

x > 0

is NO.
Since we get both 'Yes' and 'No', Statement (1) is NOT sufficient.

A statement yielding contradictory answers to a Yes/No question is insufficient.

Evaluate Statement (2) independently

By absolute value rules,

|a| = -a

holds if and only if

a \le 0

. Therefore,

|x + 2| = -(x + 2) \implies x + 2 \le 0 \implies x \le -2

.
If

x \le -2

, then

x

is strictly negative. Therefore, the answer to 'Is

x > 0

?' is ALWAYS NO.
Because Statement (2) yields a definitive 'No', Statement (2) ALONE IS SUFFICIENT.

In Data Sufficiency, a consistent and definitive 'No' response constitutes full sufficiency.

Yes/No Data Sufficiency Decision Logic: A definitive 'No' answer is just as sufficient as a definitive 'Yes' answer.

Question