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Difficulty: MediumValue vs. Yes/No Data Sufficiency Decision Logic

If pp and qq are real numbers such that p≠0p \neq 0, is p2q<0p^2 q < 0?

(1) ∣q−5∣=q−5|q - 5| = q - 5 and q≠5q \neq 5
(2) p+q=2p + q = 2

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Rephrasing the question stem reveals that because p≠0p \neq 0, p2p^2 is strictly positive; thus p2q<0p^2 q < 0 is true if and only if q<0q < 0. Statement (1) dictates that ∣q−5∣=q−5|q - 5| = q - 5 with q≠5q \neq 5, which implies q>5q > 5. Because qq is strictly positive, the answer to 'Is q<0q < 0?' is a definitive 'No'. In GMAT Data Sufficiency Yes/No questions, a statement that consistently yields a 'No' answer is sufficient. Statement (2) leaves qq undetermined (qq can be positive or negative depending on pp), so it is insufficient on its own.

Step-by-Step Solution

1
Rephrase the question stem using algebraic properties.
Since pp is a non-zero real number, p2>0p^2 > 0 for all valid pp. Dividing the inequality p2q<0p^2 q < 0 by p2p^2 yields q<0q < 0. The question simplifies to: 'Is q<0q < 0?'
Simplifying the target expression reduces cognitive complexity before evaluating the statements.
2
Evaluate Statement (1) independently.
The identity ∣k∣=k|k| = k holds if and only if k≥0k \ge 0. Thus, ∣q−5∣=q−5  ⟹  q−5≥0  ⟹  q≥5|q - 5| = q - 5 \implies q - 5 \ge 0 \implies q \ge 5. Combined with q≠5q \neq 5, we obtain q>5q > 5. If q>5q > 5, then qq is strictly positive, meaning q<0q < 0 is definitively FALSE ('No'). A definitive 'No' answer is sufficient.
In Yes/No Data Sufficiency, any statement that yields a consistent 'Yes' OR a consistent 'No' is sufficient.
3
Evaluate Statement (2) independently.
Given p+q=2p + q = 2, if p=1p = 1, then q=1>0q = 1 > 0 ('No'). If p=5p = 5, then q=−3<0q = -3 < 0 ('Yes'). Statement (2) allows both 'Yes' and 'No' outcomes.
Since Statement (2) cannot yield a single definitive Yes/No answer, it is not sufficient.

Key Concept

Value vs. Yes/No Data Sufficiency Decision Logic
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Topics

Value vs. Yes/No Data Sufficiency Decision Logic

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