If x is a real number, is |x + 4| > 2x? (1) |x - 1| < 3 (2) x^2 - x - 6 < 0 — GMAT

If $x$ is a real number, is $|x + 4| > 2x$ ?

(1) $|x - 1| < 3$
(2) $x^2 - x - 6 < 0$

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.

Rephrasing the question stem shows that

|x + 4| > 2x

is equivalent to

x < 4

. Statement (1) establishes that

-2 < x < 4

, which guarantees

x < 4

(definitive Yes). Statement (2) establishes that

-2 < x < 3

, which also guarantees

x < 4

(definitive Yes). Therefore, each statement alone is sufficient.

Rephrase the target question stem algebraically.

The inequality

|x + 4| > 2x

simplifies to

x < 4

x \ge -4

, then

x + 4 > 2x \implies x < 4

. If

x < -4

|x + 4| = -(x + 4) > 2x \implies 3x < -4 \implies x < -4/3

, which holds for all

x < -4

. Thus,

|x + 4| > 2x

is true if and only if

x < 4

Evaluate Statement (1):

|x - 1| < 3

Statement (1) yields the range

-2 < x < 4

Unpacking

|x - 1| < 3

gives

-3 < x - 1 < 3 \implies -2 < x < 4

. Since every value in

(-2, 4)

is strictly less than

4

, the answer to 'Is

x < 4

?' is a definitive 'Yes'. Thus, Statement (1) alone is sufficient.

Evaluate Statement (2):

x^2 - x - 6 < 0

Statement (2) yields the range

-2 < x < 3

Factoring gives

(x - 3)(x + 2) < 0 \implies -2 < x < 3

. Since every value in

(-2, 3)

is strictly less than

4

, the answer to 'Is

x < 4

?' is a definitive 'Yes'. Thus, Statement (2) alone is sufficient.

Rephrasing absolute value inequalities in Data Sufficiency Yes/No questions

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