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Difficulty: HardQuestion Stem Simplification and Target Rephrasing

If xx and yy are real numbers with x≠0x \neq 0, is x4−y4x3+xy2>x−y\frac{x^4 - y^4}{x^3 + x y^2} > x - y?

(1) y=2xy = 2x
(2) xy>0xy > 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. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. Statements (1) and (2) TOGETHER are NOT sufficient.Answer

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.
Simplifying the numerator (x4−y4)=(x2−y2)(x2+y2)(x^4 - y^4) = (x^2 - y^2)(x^2 + y^2) and denominator x3+xy2=x(x2+y2)x^3 + xy^2 = x(x^2 + y^2) reduces the left side of the inequality to x2−y2x\frac{x^2 - y^2}{x}. Rephrasing the question x2−y2x>x−y\frac{x^2 - y^2}{x} > x - y shows that it is equivalent to asking whether y(x−y)x>0\frac{y(x - y)}{x} > 0.

Under Statement (1), substituting y=2xy = 2x transforms the target into asking 'Is x<0x < 0?'. Because Statement (1) gives no information about the sign of xx, it is insufficient.

Under Statement (2), xy>0xy > 0 means xx and yy share the same sign, but testing values shows both 'Yes' and 'No' are possible (x=2,y=1x=2, y=1 yields Yes, whereas x=1,y=2x=1, y=2 yields No). Thus Statement (2) is insufficient.

Combining both statements gives y=2xy = 2x and xy=2x2>0xy = 2x^2 > 0, which holds true for all non-zero real numbers regardless of sign. Because xx can still be positive or negative, the question 'Is x<0x < 0?' cannot be answered. Hence, both statements together are NOT sufficient, making the option stating that statements (1) and (2) together are not sufficient the correct choice.

Step-by-Step Solution

1
Simplify the algebraic expression on the left-hand side of the target inequality.
x4−y4x3+xy2=(x2−y2)(x2+y2)x(x2+y2)=x2−y2x\frac{x^4 - y^4}{x^3 + xy^2} = \frac{(x^2 - y^2)(x^2 + y^2)}{x(x^2 + y^2)} = \frac{x^2 - y^2}{x} since x2+y2>0x^2 + y^2 > 0 for all non-zero real xx.
Factoring the numerator as a difference of squares and factoring out xx in the denominator allows canceling the non-zero common factor (x2+y2)(x^2 + y^2).
2
Rephrase the target inequality in terms of simplified components.
The target question 'Is x2−y2x>x−y\frac{x^2 - y^2}{x} > x - y?' subtracts to x2−y2−x(x−y)x>0  ⟹  y(x−y)x>0\frac{x^2 - y^2 - x(x - y)}{x} > 0 \implies \frac{y(x - y)}{x} > 0.
Simplifying the target question establishes the precise algebraic condition that must be evaluated by the statements.
3
Evaluate Statement (1): y=2xy = 2x.
Substituting y=2xy = 2x into the simplified target yields 2x(x−2x)x>0  ⟹  −2x2x>0  ⟹  −2x>0  ⟹  x<0\frac{2x(x - 2x)}{x} > 0 \implies \frac{-2x^2}{x} > 0 \implies -2x > 0 \implies x < 0.
Statement (1) reduces the question to 'Is x<0x < 0?'. Without information about the sign of xx, this statement alone is NOT sufficient.
4
Evaluate Statement (2): xy>0xy > 0.
If x=2,y=1x = 2, y = 1, 1(2−1)2=0.5>0\frac{1(2-1)}{2} = 0.5 > 0 (Yes). If x=1,y=2x = 1, y = 2, 2(1−2)1=−2≯0\frac{2(1-2)}{1} = -2 \ngtr 0 (No).
Knowing that xx and yy have the same sign does not determine relative magnitudes, so Statement (2) alone is NOT sufficient.
5
Evaluate Statements (1) and (2) together.
Combining y=2xy = 2x and xy>0xy > 0 gives x(2x)=2x2>0x(2x) = 2x^2 > 0, which is true for all non-zero real xx. If x=1,y=2x = 1, y = 2, the target gives 'No'; if x=−1,y=−2x = -1, y = -2, the target gives 'Yes'.
Combining both statements still does not determine whether x<0x < 0 or x>0x > 0, so both statements together are NOT sufficient.

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

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