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Difficulty: Very hardInequalities, Absolute Values, and Number Ranges in Data Sufficiency

If aa and bb are non-zero real numbers, is ∣a+b∣>∣a−b∣|a + b| > |a - b|?

(1) ∣a+2b∣<∣a−2b∣|a + 2b| < |a - 2b|
(2) ∣a∣b+∣b∣a<0\frac{|a|}{b} + \frac{|b|}{a} < 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.
Rephrasing the question stem by squaring both sides gives ∣a+b∣2>∣a−b∣2  ⟺  (a+b)2>(a−b)2  ⟺  4ab>0  ⟺  ab>0|a + b|^2 > |a - b|^2 \iff (a + b)^2 > (a - b)^2 \iff 4ab > 0 \iff ab > 0. Statement (1) simplifies similarly by squaring both sides to (a+2b)2<(a−2b)2  ⟺  8ab<0  ⟺  ab<0(a + 2b)^2 < (a - 2b)^2 \iff 8ab < 0 \iff ab < 0. Since ab<0ab < 0 guarantees that ab>0ab > 0 is false, Statement (1) produces a definitive 'No' and is therefore sufficient. Statement (2) is satisfied when both aa and bb are negative (giving ab>0ab > 0, a 'Yes' answer) as well as when aa and bb have opposite signs with ∣a∣>∣b∣|a| > |b| (giving ab<0ab < 0, a 'No' answer). Because Statement (2) permits both outcomes, it is insufficient.

Step-by-Step Solution

1
Rephrase the question stem target using algebraic properties of absolute values.
Squaring both non-negative sides of ∣a+b∣>∣a−b∣|a + b| > |a - b| yields (a+b)2>(a−b)2  ⟹  a2+2ab+b2>a2−2ab+b2  ⟹  4ab>0  ⟹  ab>0(a + b)^2 > (a - b)^2 \implies a^2 + 2ab + b^2 > a^2 - 2ab + b^2 \implies 4ab > 0 \implies ab > 0. The question asks: 'Is ab>0ab > 0?'
Simplifying the stem target into a fundamental sign constraint (ab>0ab > 0) prevents algebraic errors during statement evaluation.
2
Evaluate Statement (1): ∣a+2b∣<∣a−2b∣|a + 2b| < |a - 2b|.
Squaring both sides yields (a+2b)2<(a−2b)2  ⟹  a2+4ab+4b2<a2−4ab+4b2  ⟹  8ab<0  ⟹  ab<0(a + 2b)^2 < (a - 2b)^2 \implies a^2 + 4ab + 4b^2 < a^2 - 4ab + 4b^2 \implies 8ab < 0 \implies ab < 0.
Since ab<0ab < 0, abab cannot be greater than 00. This yields a definitive 'No' to the question 'Is ab>0ab > 0?'. A definitive 'No' is sufficient.
3
Evaluate Statement (2): ∣a∣b+∣b∣a<0\frac{|a|}{b} + \frac{|b|}{a} < 0.
Combine the fractions: a∣a∣+b∣b∣ab<0\frac{a|a| + b|b|}{ab} < 0. Test specific cases:
- Case A: a=−2,b=−2  ⟹  ∣−2∣−2+∣−2∣−2=−2<0a = -2, b = -2 \implies \frac{|-2|}{-2} + \frac{|-2|}{-2} = -2 < 0. Here ab=4>0ab = 4 > 0 (Answer: YES).
- Case B: a=3,b=−1  ⟹  ∣3∣−1+∣−1∣3=−3+13=−83<0a = 3, b = -1 \implies \frac{|3|}{-1} + \frac{|-1|}{3} = -3 + \frac{1}{3} = -\frac{8}{3} < 0. Here ab=−3<0ab = -3 < 0 (Answer: NO).
Since Statement (2) allows both a 'Yes' and a 'No' response to whether ab>0ab > 0, Statement (2) alone is not sufficient.

Key Concept

Data Sufficiency Yes/No decision logic combined with distance and squaring properties of absolute value inequalities.
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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