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Zorluk: Çok zorQuestion Stem Simplification and Target Rephrasing

If xx and yy are real numbers such that x≠yx \neq y and x+y≠0x + y \neq 0, is x3−y3x2−y2>x+y\frac{x^3 - y^3}{x^2 - y^2} > x + y?

(1) x>y>0x > y > 0

(2) ∣x+y∣<∣x−y∣|x + y| < |x - y|

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Cevap
  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.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Factoring the numerator and denominator of the target expression transforms the question 'Is x3−y3x2−y2>x+y\frac{x^3 - y^3}{x^2 - y^2} > x + y?' into the simplified target 'Is xyx+y<0\frac{xy}{x + y} < 0?'. Under Statement (1), both xx and yy are positive, so xy>0xy > 0 and x+y>0x + y > 0, making xyx+y>0\frac{xy}{x + y} > 0. This conclusively answers the target question with a definitive 'No', proving Statement (1) is sufficient alone. Statement (2) simplifies to xy<0xy < 0, which allows x+yx + y to be positive or negative, resulting in both 'Yes' and 'No' possibilities, rendering Statement (2) insufficient.

Adım Adım Çözüm

1
Rephrase the question target algebraically before evaluating the statements.
Simplifying x3−y3x2−y2\frac{x^3 - y^3}{x^2 - y^2} by factoring numerator and denominator yields (x−y)(x2+xy+y2)(x−y)(x+y)=x2+xy+y2x+y\frac{(x - y)(x^2 + xy + y^2)}{(x - y)(x + y)} = \frac{x^2 + xy + y^2}{x + y}. The inequality x2+xy+y2x+y>x+y\frac{x^2 + xy + y^2}{x + y} > x + y simplifies to x2+xy+y2−(x+y)2x+y>0\frac{x^2 + xy + y^2 - (x + y)^2}{x + y} > 0, which reduces to −xyx+y>0\frac{-xy}{x + y} > 0, or equivalently xyx+y<0\frac{xy}{x + y} < 0.
Target rephrasing reveals that the question is asking whether xyxy and x+yx + y have opposite signs.
2
Evaluate Statement (1): x>y>0x > y > 0.
Since x>0x > 0 and y>0y > 0, xy>0xy > 0 and x+y>0x + y > 0. Consequently, xyx+y>0\frac{xy}{x + y} > 0. This yields a definitive 'No' to the rephrased question 'Is xyx+y<0\frac{xy}{x + y} < 0?'. A definitive 'No' is sufficient.
A statement that guarantees a definitive 'No' answer to a Yes/No question is sufficient.
3
Evaluate Statement (2): ∣x+y∣<∣x−y∣|x + y| < |x - y|.
Squaring both sides gives (x+y)2<(x−y)2  ⟹  x2+2xy+y2<x2−2xy+y2  ⟹  4xy<0  ⟹  xy<0(x + y)^2 < (x - y)^2 \implies x^2 + 2xy + y^2 < x^2 - 2xy + y^2 \implies 4xy < 0 \implies xy < 0. However, knowing xy<0xy < 0 leaves the sign of x+yx + y undetermined. For example, if x=3,y=−1x = 3, y = -1, then x+y=2>0x + y = 2 > 0 and xyx+y=−1.5<0\frac{xy}{x+y} = -1.5 < 0 (Yes). If x=1,y=−3x = 1, y = -3, then x+y=−2<0x + y = -2 < 0 and xyx+y=1.5>0\frac{xy}{x+y} = 1.5 > 0 (No).
Since the outcome can be either 'Yes' or 'No', Statement (2) alone is not sufficient.

Anahtar Kavram

Data Sufficiency Target Simplification and Yes/No Decision Logic
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