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Difficulty: Very hardValue vs. Yes/No Data Sufficiency Decision Logic

If xx and yy are non-zero real numbers, is xy+yx<2\frac{x}{y} + \frac{y}{x} < 2?

(1) (x+y)2>x2+y2(x + y)^2 > x^2 + y^2
(2) x3y2<0x^3 y^2 < 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 to answer the question with a definitive 'No', but statement (2) alone is not sufficient.
The correct answer identifies that Statement (1) provides sufficient information by demonstrating that the expression is always greater than or equal to 2, yielding a definitive 'No' to the question stem. Statement (2) leaves the sign of one variable unspecified, allowing both 'Yes' and 'No' answers, making it insufficient.

Step-by-Step Solution

1
Rephrase the question stem using algebraic identities.
For any non-zero real numbers xx and yy, xy+yx=x2+y2xy\frac{x}{y} + \frac{y}{x} = \frac{x^2 + y^2}{xy}. Note that (x−y)2=x2−2xy+y2≥0(x - y)^2 = x^2 - 2xy + y^2 \ge 0, which implies x2+y2≥2xyx^2 + y^2 \ge 2xy. If xy>0xy > 0, dividing by xyxy gives x2+y2xy≥2\frac{x^2 + y^2}{xy} \ge 2 (so the inequality is FALSE, answer NO). If xy<0xy < 0, then xy<0\frac{x}{y} < 0 and yx<0\frac{y}{x} < 0, making their sum negative and thus strictly less than 2 (answer YES). Therefore, determining whether xy>0xy > 0 or xy<0xy < 0 answers the Yes/No question.
Simplifying the target expression establishes the exact algebraic condition (xy>0xy > 0 vs xy<0xy < 0) needed to answer the question.
2
Evaluate Statement (1): (x+y)2>x2+y2(x + y)^2 > x^2 + y^2.
Expanding the left side yields x2+2xy+y2>x2+y2x^2 + 2xy + y^2 > x^2 + y^2, which simplifies to 2xy>02xy > 0, or xy>0xy > 0. As shown in Step 1, when xy>0xy > 0, xy+yx≥2\frac{x}{y} + \frac{y}{x} \ge 2, meaning the expression is NEVER less than 2. This gives a definitive answer of 'NO' to the stem question.
In GMAT Data Sufficiency Yes/No questions, a statement that establishes a definitive 'NO' is fully SUFFICIENT.
3
Evaluate Statement (2): x3y2<0x^3 y^2 < 0.
Since y≠0y \neq 0, y2>0y^2 > 0 for all real yy. Dividing by y2y^2 yields x3<0x^3 < 0, which means x<0x < 0. However, Statement (2) provides no information about the sign of yy. If y<0y < 0, then xy>0xy > 0 and the answer is 'NO'. If y>0y > 0, then xy<0xy < 0 and the answer is 'YES'. Since both 'YES' and 'NO' are possible, Statement (2) is NOT sufficient.
A statement that permits both 'Yes' and 'No' outcomes fails to provide a conclusive determination.

Key Concept

In Yes/No Data Sufficiency questions, sufficiency requires a consistent, definitive answer—either a guaranteed 'Yes' or a guaranteed 'No'. Proving that a statement is conclusively 'No' satisfies sufficiency.
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Topics

Value vs. Yes/No Data Sufficiency Decision Logic

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