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Difficulty: MediumQuestion Stem Simplification and Target Rephrasing

If pp and qq are non-zero real numbers such that p≠qp \neq q, is p3q−pq3p−q>0\frac{p^3 q - p q^3}{p - q} > 0?

(1) pq>0pq > 0
(2) p+q>0p + q > 0

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Answer
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Factoring the numerator gives p3q−pq3=pq(p−q)(p+q)p^3 q - p q^3 = pq(p-q)(p+q). Since p≠qp \neq q, dividing by (p−q)(p-q) yields pq(p+q)pq(p+q). To determine whether pq(p+q)>0pq(p+q) > 0, we need the sign of the product pq(p+q)pq(p+q). Statement (1) tells us pq>0pq > 0, which is insufficient by itself because p+qp+q could be positive or negative. Statement (2) tells us p+q>0p+q > 0, which is insufficient by itself because pqpq could be positive or negative. Together, both terms pqpq and p+qp+q are positive, so their product must be positive, providing a definitive 'Yes'.

Step-by-Step Solution

1
Rephrase the target question stem algebraically by factoring the numerator.
p3q−pq3p−q=pq(p2−q2)p−q=pq(p−q)(p+q)p−q=pq(p+q)\frac{p^3 q - p q^3}{p - q} = \frac{p q (p^2 - q^2)}{p - q} = \frac{p q (p - q)(p + q)}{p - q} = pq(p + q). The target question simplifies to: 'Is pq(p+q)>0pq(p+q) > 0?'
Simplifying the stem isolates the specific product whose sign determines sufficiency.
2
Evaluate Statement (1): pq>0pq > 0.
If p=1,q=1p = 1, q = 1, then pq=1>0pq = 1 > 0 and pq(p+q)=1(2)=2>0pq(p+q) = 1(2) = 2 > 0 (Yes). If p=−1,q=−1p = -1, q = -1, then pq=1>0pq = 1 > 0 and pq(p+q)=1(−2)=−2<0pq(p+q) = 1(-2) = -2 < 0 (No).
Knowing pq>0pq > 0 leaves the sign of p+qp+q unknown, so Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2): p+q>0p + q > 0.
If p=2,q=1p = 2, q = 1, then p+q=3>0p+q = 3 > 0 and pq(p+q)=2(3)=6>0pq(p+q) = 2(3) = 6 > 0 (Yes). If p=3,q=−1p = 3, q = -1, then p+q=2>0p+q = 2 > 0 and pq(p+q)=(−3)(2)=−6<0pq(p+q) = (-3)(2) = -6 < 0 (No).
Knowing p+q>0p+q > 0 leaves the sign of pqpq unknown, so Statement (2) alone is NOT sufficient.
4
Evaluate Statement (1) and Statement (2) together.
From Statement (1), pq>0pq > 0. From Statement (2), p+q>0p+q > 0. Therefore, pq(p+q)>0pq(p+q) > 0 as it is the product of two strictly positive real numbers.
Combining both statements provides a definitive 'Yes' answer to the simplified target question.

Key Concept

Question Stem Simplification and Target Rephrasing
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Question Stem Simplification and Target Rephrasing

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