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

If aa and bb are real numbers, is a2−b2>0a^2 - b^2 > 0?

(1) ∣a∣<b|a| < b
(2) a+b>0a + b > 0

  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.
Statement (1) establishes that ∣a∣<b|a| < b, which implies b>0b > 0 and a2<b2a^2 < b^2. This guarantees that a2−b2<0a^2 - b^2 < 0, leading to a definitive 'No' answer to the question 'Is a2−b2>0a^2 - b^2 > 0?'. In GMAT Data Sufficiency Yes/No questions, a statement that consistently leads to a 'No' answer is sufficient. Statement (2) provides a+b>0a + b > 0, which allows for cases where a2−b2>0a^2 - b^2 > 0 (e.g., a=3,b=1a=3, b=1) and cases where a2−b2<0a^2 - b^2 < 0 (e.g., a=1,b=3a=1, b=3), making it insufficient.

Step-by-Step Solution

1
Analyze the target question stem
The question asks whether a2−b2>0a^2 - b^2 > 0, which is equivalent to asking if ∣a∣>∣b∣|a| > |b| or if a2>b2a^2 > b^2. This is a Yes/No Data Sufficiency question.
Establishing what condition constitutes a 'Yes' versus a 'No' is essential before evaluating individual statements.
2
Evaluate Statement (1): ∣a∣<b|a| < b
Since ∣a∣≥0|a| \ge 0, it follows that b>0b > 0. Squaring both non-negative sides of ∣a∣<b|a| < b gives ∣a∣2<b2|a|^2 < b^2, which simplifies to a2<b2a^2 < b^2. Subtracting b2b^2 yields a2−b2<0a^2 - b^2 < 0.
Because a2−b2<0a^2 - b^2 < 0 for all values satisfying Statement (1), the answer to 'Is a2−b2>0a^2 - b^2 > 0?' is a definitive 'No'. In Data Sufficiency, a definitive 'No' means Statement (1) is ALONE sufficient.
3
Evaluate Statement (2): a+b>0a + b > 0
If a=3a = 3 and b=1b = 1, then a+b=4>0a + b = 4 > 0 and a2−b2=9−1=8>0a^2 - b^2 = 9 - 1 = 8 > 0 (Yes). If a=1a = 1 and b=3b = 3, then a+b=4>0a + b = 4 > 0 and a2−b2=1−9=−8<0a^2 - b^2 = 1 - 9 = -8 < 0 (No).
Since Statement (2) yields both 'Yes' and 'No' responses depending on the specific values chosen, Statement (2) is NOT sufficient.
4
Determine the final Data Sufficiency decision logic
Statement (1) alone is sufficient to yield a definitive 'No' answer, while Statement (2) alone is not sufficient.
In GMAT Data Sufficiency Yes/No questions, any statement that provides a single, conclusive answer (whether 'Yes' or 'No') is sufficient.

Key Concept

Definitive No Sufficiency in Yes/No Data Sufficiency
Estimated Time:2m 0s
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Topics

Value vs. Yes/No Data Sufficiency Decision Logic

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