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Difficulty: MediumInequalities, Absolute Values, and Number Ranges in Data Sufficiency

If pp and qq are non-zero real numbers, is ∣p+q∣>∣p−q∣|p + q| > |p - q|?

(1) ∣p+2q∣<∣p−2q∣|p + 2q| < |p - 2q|
(2) p−q>0p - q > 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.
The correct response identifies Statement (1) as sufficient and Statement (2) as insufficient. Rephrasing the question stem by squaring both sides of ∣p+q∣>∣p−q∣|p + q| > |p - q| yields 4pq>04pq > 0, or pq>0pq > 0. Statement (1) expands and simplifies to 8pq<08pq < 0, meaning pq<0pq < 0. This provides a definitive 'No' answer to the question 'Is pq>0pq > 0?', which constitutes full sufficiency. Statement (2) merely tells us p>qp > q, which permits pp and qq to both be positive (pq>0pq > 0) or to have opposite signs (pq<0pq < 0), rendering Statement (2) insufficient.

Step-by-Step Solution

1
Rephrase the question stem target geometrically or algebraically.
Squaring both non-negative sides of ∣p+q∣>∣p−q∣|p + q| > |p - q| gives (p+q)2>(p−q)2  ⟹  p2+2pq+q2>p2−2pq+q2  ⟹  4pq>0  ⟹  pq>0(p + q)^2 > (p - q)^2 \implies p^2 + 2pq + q^2 > p^2 - 2pq + q^2 \implies 4pq > 0 \implies pq > 0. The question asks: Is pq>0pq > 0?
Simplifying the question stem simplifies evaluation of both statements by focusing on whether pp and qq have the same sign.
2
Evaluate Statement (1): ∣p+2q∣<∣p−2q∣|p + 2q| < |p - 2q|.
Squaring both non-negative sides gives (p+2q)2<(p−2q)2  ⟹  p2+4pq+4q2<p2−4pq+4q2  ⟹  8pq<0  ⟹  pq<0(p + 2q)^2 < (p - 2q)^2 \implies p^2 + 4pq + 4q^2 < p^2 - 4pq + 4q^2 \implies 8pq < 0 \implies pq < 0.
Since pq<0pq < 0, we can answer the rephrased target question ('Is pq>0pq > 0?') with a definitive 'NO'. In Data Sufficiency, a definitive 'No' means the statement is sufficient.
3
Evaluate Statement (2): p−q>0p - q > 0.
If p=3p = 3 and q=1q = 1, then p−q=2>0p - q = 2 > 0 and pq=3>0pq = 3 > 0 (Yes). If p=1p = 1 and q=−2q = -2, then p−q=3>0p - q = 3 > 0 and pq=−2<0pq = -2 < 0 (No).
Because Statement (2) allows both 'Yes' and 'No' outcomes, it is not sufficient.

Key Concept

Rephrasing absolute value distance inequalities and recognizing that a definitive 'No' response in Yes/No Data Sufficiency constitutes sufficiency.
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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