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Difficulty: 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−1∣<∣b−1∣|a - 1| < |b - 1|

Which of the following choices correctly describes the sufficiency of the statements?

  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.
The option stating that Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient is correct. Squaring both sides of the question stem inequality ∣a−b∣>∣a+b∣|a - b| > |a + b| yields a2−2ab+b2>a2+2ab+b2a^2 - 2ab + b^2 > a^2 + 2ab + b^2, which simplifies directly to ab<0ab < 0. Statement (1) states ∣a+2b∣<∣a−2b∣|a + 2b| < |a - 2b|. Squaring both sides yields a2+4ab+4b2<a2−4ab+4b2a^2 + 4ab + 4b^2 < a^2 - 4ab + 4b^2, which simplifies to 8ab<08ab < 0, or ab<0ab < 0. This gives a definitive YES to the question, so Statement (1) is sufficient. Statement (2) compares the distances of aa and bb from 1 on the number line. Choosing a=0.5a = 0.5 and b=−2b = -2 satisfies ∣0.5−1∣<∣−2−1∣|0.5 - 1| < |-2 - 1| and gives ab=−1<0ab = -1 < 0 (YES), whereas choosing a=0.5a = 0.5 and b=3b = 3 satisfies ∣0.5−1∣<∣3−1∣|0.5 - 1| < |3 - 1| and gives ab=1.5>0ab = 1.5 > 0 (NO). Thus, Statement (2) is insufficient.

Step-by-Step Solution

1
Rephrase the question stem target using absolute value properties.
Since both sides of ∣a−b∣>∣a+b∣|a - b| > |a + b| are non-negative, square both sides: (a−b)2>(a+b)2  ⟹  a2−2ab+b2>a2+2ab+b2  ⟹  −2ab>2ab  ⟹  4ab<0  ⟹  ab<0(a - b)^2 > (a + b)^2 \implies a^2 - 2ab + b^2 > a^2 + 2ab + b^2 \implies -2ab > 2ab \implies 4ab < 0 \implies ab < 0. Thus, the question asks: 'Is ab<0ab < 0?' (Do aa and bb have opposite signs?)
Simplifying the target inequality converts an absolute value relationship into a clear sign constraint on the product abab.
2
Evaluate Statement (1): ∣a+2b∣<∣a−2b∣|a + 2b| < |a - 2b|.
Square both sides: (a+2b)2<(a−2b)2  ⟹  a2+4ab+4b2<a2−4ab+4b2  ⟹  4ab<−4ab  ⟹  8ab<0  ⟹  ab<0(a + 2b)^2 < (a - 2b)^2 \implies a^2 + 4ab + 4b^2 < a^2 - 4ab + 4b^2 \implies 4ab < -4ab \implies 8ab < 0 \implies ab < 0. This provides a definitive YES to the question 'Is ab<0ab < 0?'
Statement (1) uniquely guarantees that aa and bb have opposite signs, making Statement (1) ALONE sufficient.
3
Evaluate Statement (2): ∣a−1∣<∣b−1∣|a - 1| < |b - 1|.
Test numbers: Case 1: If a=0.5a = 0.5 and b=−2b = -2, then ∣0.5−1∣=0.5|0.5 - 1| = 0.5 and ∣−2−1∣=3|-2 - 1| = 3. Since 0.5<30.5 < 3, Statement (2) holds, and ab=(0.5)(−2)=−1<0ab = (0.5)(-2) = -1 < 0 (YES). Case 2: If a=0.5a = 0.5 and b=3b = 3, then ∣0.5−1∣=0.5|0.5 - 1| = 0.5 and ∣3−1∣=2|3 - 1| = 2. Since 0.5<20.5 < 2, Statement (2) holds, and ab=(0.5)(3)=1.5>0ab = (0.5)(3) = 1.5 > 0 (NO).
Because Statement (2) permits both ab<0ab < 0 and ab>0ab > 0, it cannot determine whether ab<0ab < 0. Statement (2) ALONE is insufficient.

Key Concept

Properties of Absolute Value Inequalities and Rephrased Data Sufficiency Targets
Estimated Time:2m 0s
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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