If x is a real number, is |2x - 3| \le 5? (1) |x - 1| \le 2 (2) x^2 - 3x - 4 \le 0 — GMAT

If $x$ is a real number, is $|2x - 3| \le 5$ ?

(1) $|x - 1| \le 2$
(2) $x^2 - 3x - 4 \le 0$

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 target inequality

|2x - 3| \le 5

gives

-1 \le x \le 4

. Statement (1) yields

-1 \le x \le 3

, which is completely contained within

[-1, 4]

and therefore guarantees a definitive 'Yes'. Statement (2) yields

-1 \le x \le 4

, which matches the condition directly and also gives a definitive 'Yes'. Hence, each statement alone is sufficient.

Rephrase the question stem target inequality algebraically.

-5 \le 2x - 3 \le 5 \implies -2 \le 2x \le 8 \implies -1 \le x \le 4

. Target Question: Is

-1 \le x \le 4

Simplifying the question stem converts an absolute value inequality into a clear number range for

x

Evaluate Statement (1):

|x - 1| \le 2

-2 \le x - 1 \le 2 \implies -1 \le x \le 3

. Since every value in

[-1, 3]

is also in

[-1, 4]

, the answer to 'Is

-1 \le x \le 4

?' is a definitive YES.

If a statement's allowed range is a subset of the target range, it guarantees a definitive 'Yes' answer.

Evaluate Statement (2):

x^2 - 3x - 4 \le 0

Factor the quadratic:

(x - 4)(x + 1) \le 0 \implies -1 \le x \le 4

. This matches the target range exactly, yielding a definitive YES.

Matching the target inequality range provides a definitive 'Yes' answer.

Data Sufficiency Yes/No decision logic for absolute value range constraints and subset range implications.

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