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Difficulty: MediumQuestion Stem Simplification and Target Rephrasing

For all positive real numbers pp and qq, the Data Sufficiency Yes/No target question "Is p2+q22pq>1\frac{p^2 + q^2}{2pq} > 1?" is algebraically equivalent to the simplified target question "Is p≠qp \neq q?"

Answer: Answer

Answer

True
The statement is true because clearing the positive denominator 2pq2pq and rearranging terms converts the target inequality p2+q22pq>1\frac{p^2 + q^2}{2pq} > 1 into (p−q)2>0(p - q)^2 > 0. A real squared quantity is strictly positive if and only if its base is non-zero, making the target equivalent to "Is p≠qp \neq q?"

Step-by-Step Solution

1
Multiply both sides of the target inequality p2+q22pq>1\frac{p^2 + q^2}{2pq} > 1 by 2pq2pq.
p2+q2>2pqp^2 + q^2 > 2pq, with the direction of the inequality preserved since p>0p > 0 and q>0q > 0 implies 2pq>02pq > 0.
Clearing positive denominators simplifies fractional inequalities without changing the inequality sign.
2
Subtract 2pq2pq from both sides of the inequality.
p2−2pq+q2>0p^2 - 2pq + q^2 > 0.
Grouping all non-zero terms on one side sets up a recognizable quadratic form.
3
Factor the quadratic expression as a perfect square.
(p−q)2>0(p - q)^2 > 0.
Recognizing p2−2pq+q2p^2 - 2pq + q^2 as the expansion of (p−q)2(p - q)^2 isolates the core variable relationship.
4
Analyze the conditions required for (p−q)2>0(p - q)^2 > 0 to hold for real numbers.
The square of any real number is always non-negative, and it is strictly greater than zero if and only if the base is non-zero (p−q≠0p - q \neq 0, which means p≠qp \neq q).
Simplifying the logical condition reveals the exact target question needed for Data Sufficiency evaluation.

Key Concept

Target Rephrasing via Perfect Square Inequalities
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Topics

Question Stem Simplification and Target Rephrasing

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