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Difficulty: MediumInequalities, 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) a3b2<0a^3 b^2 < 0
(2) ∣a−b∣>∣a+b∣|a - b| > |a + b|

Which of the following describes the sufficiency of the statements?

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Answer
  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 (2) ALONE is sufficient, but statement (1) alone is not sufficient.
The correct response identifies that statement (2) alone provides enough information to answer the question, whereas statement (1) does not. Rephrasing the stem ∣a+b∣<∣a∣+∣b∣|a + b| < |a| + |b| reveals it is true if and only if aa and bb have opposite signs (ab<0ab < 0). Statement (1) only confirms that a<0a < 0, leaving the sign of bb unknown. Statement (2) can be squared to yield ab<0ab < 0, which conclusively answers the question with a 'Yes'.

Step-by-Step Solution

1
Rephrase the question stem
The inequality ∣a+b∣<∣a∣+∣b∣|a + b| < |a| + |b| holds if and only if aa and bb have opposite signs (i.e., ab<0ab < 0). If aa and bb have the same sign or if either is zero, ∣a+b∣=∣a∣+∣b∣|a + b| = |a| + |b|. Thus, the target question is: 'Is ab<0ab < 0?'
Simplifying absolute value properties reduces the stem to a straightforward sign comparison.
2
Evaluate Statement (1): a3b2<0a^3 b^2 < 0
Since b≠0b \neq 0, b2>0b^2 > 0 for all real numbers. Thus, a3b2<0a^3 b^2 < 0 implies a3<0a^3 < 0, which means a<0a < 0. However, bb can be either positive or negative. If b>0b > 0, then ab<0ab < 0 (YES). If b<0b < 0, then ab>0ab > 0 (NO).
Since statement (1) allows both a YES and a NO answer, Statement (1) ALONE is NOT sufficient.
3
Evaluate Statement (2): ∣a−b∣>∣a+b∣|a - b| > |a + b|
Square both sides (valid since both sides are non-negative): (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.
Statement (2) directly proves that ab<0ab < 0, yielding a definitive YES to the question stem. Thus, Statement (2) ALONE is SUFFICIENT.

Key Concept

Absolute Value Distance and Sign Properties
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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