For all non-zero real numbers a and b with |a| \neq |b|, the Data Sufficiency Yes/No targe… — GMAT

For all non-zero real numbers $a$ and $b$ with $|a| \neq |b|$ , the Data Sufficiency Yes/No target question "Is $\frac{a^2 - b^2}{ab} > 0$ ?" is algebraically equivalent to asking whether $a$ and $b$ have the same sign when $|a| > |b|$ , or different signs when $|a| < |b|$ .

Answer: Answer

Answer

True. The rephrased question accurately state the necessary and sufficient conditions for the fraction

\frac{a^2 - b^2}{ab}

to be positive.

The quotient

\frac{a^2 - b^2}{ab}

is positive when both

a^2 - b^2

and

ab

are positive, or when both are negative. If

a

and

b

have the same sign (

ab > 0

), then

a^2 - b^2 > 0

, which simplifies to

a^2 > b^2

|a| > |b|

. If

a

and

b

have different signs (

ab < 0

), then

a^2 - b^2 < 0

, which simplifies to

a^2 < b^2

|a| < |b|

. Hence, the statement correctly describes the target simplification.

Step-by-Step Solution

Set up the quotient inequality condition

The fraction

\frac{a^2 - b^2}{ab} > 0

holds when numerator

a^2 - b^2

and denominator

ab

have matching signs.

For any real fraction

\frac{N}{D} > 0

, either (

N > 0

and

D > 0

) or (

N < 0

and

D < 0

Analyze Case 1 where denominator

ab > 0

a

and

b

have the same sign. In this case,

\frac{a^2 - b^2}{ab} > 0 \implies a^2 - b^2 > 0 \implies |a| > |b|

When

ab > 0

, multiplying both sides of the inequality by

ab

preserves the inequality sign.

Analyze Case 2 where denominator

ab < 0

a

and

b

have different signs. In this case,

\frac{a^2 - b^2}{ab} > 0 \implies a^2 - b^2 < 0 \implies |a| < |b|

When

ab < 0

, multiplying both sides of the inequality by

ab

flips the inequality direction.

Combine the cases to evaluate the target rephrasing

The target question simplifies to: "Do

a

and

b

have the same sign with

|a| > |b|

, or opposite signs with

|a| < |b|

Both cases together form the complete rephrased condition for the original statement to be true.

Key Concept

Algebraic Rephrasing of Quotients involving Absolute Values and Variable Signs

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