For all distinct non-zero real numbers x and y, the Data Sufficiency Yes/No target questio… — GMAT

For all distinct non-zero real numbers $x$ and $y$ , the Data Sufficiency Yes/No target question "Is $\frac{x|y| - y|x|}{x - y} > 0$ ?" is algebraically equivalent to the simplified target question "Is $xy < 0$ ?"

Answer: Answer

Answer

The statement is true. Simplifying the target question by considering the relative signs of

x

and

y

proves that the expression is strictly positive if and only if

x

and

y

have opposite signs (

xy < 0

Rephrasing a Data Sufficiency target question requires finding a simpler, logically equivalent condition. Testing the two distinct operational domains (

xy > 0

vs.

xy < 0

) demonstrates that the given algebraic fraction yields a positive result if and only if

x

and

y

carry opposite signs (

xy < 0

). Therefore, the proposed rephrasing is completely accurate and equivalent.

Step-by-Step Solution

Analyze the expression when

x

and

y

have the same sign (

xy > 0

x > 0, y > 0

, then

|x| = x

and

|y| = y

, so

x|y| - y|x| = xy - yx = 0

. If

x < 0, y < 0

, then

|x| = -x

and

|y| = -y

, so

x|y| - y|x| = x(-y) - y(-x) = -xy + xy = 0

. Thus, when

xy > 0

\frac{x|y| - y|x|}{x - y} = 0

, yielding a 'No' to the question 'Is the expression

> 0

?'.

Determining the behavior of absolute value expressions under identical signs simplifies the numerator to zero.

Analyze the expression when

x

and

y

have opposite signs (

xy < 0

Case A: If

x > 0

and

y < 0

, then

|x| = x

and

|y| = -y

. The numerator is

x(-y) - y(x) = -2xy > 0

. The denominator is

x - y > 0

. The quotient is

\frac{\text{positive}}{\text{positive}} > 0

.
Case B: If

x < 0

and

y > 0

, then

|x| = -x

and

|y| = y

. The numerator is

x(y) - y(-x) = 2xy < 0

. The denominator is

x - y < 0

. The quotient is

\frac{\text{negative}}{\text{negative}} > 0

Evaluating absolute values under opposite signs demonstrates that the numerator and denominator always have matching signs.

Compare the conditions for a 'Yes' answer.

The target expression is strictly positive if and only if

x

and

y

have opposite signs, which is defined by the inequality

xy < 0

Establishing biconditional equivalence confirms that rephrasing the target stem to 'Is

xy < 0

?' preserves all logical outcomes.

Key Concept

Data Sufficiency target rephrasing using piecewise definition of absolute value and sign analysis.

Question

Answer

Step-by-Step Solution

Key Concept

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