For all real numbers x such that x \neq 0 and x \neq 1, the Data Sufficiency Yes/No target… — GMAT

For all real numbers $x$ such that $x \neq 0$ and $x \neq 1$ , the Data Sufficiency Yes/No target question "Is $\frac{x^2 - x}{|x - 1|} > 0$ ?" is algebraically equivalent to asking "Is $x > 0$ ?".

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The statement is False.

The statement is False because simplifying the target inequality

\frac{x(x - 1)}{|x - 1|} > 0

yields

x < 0

x > 1

, which is not equivalent to

x > 0

Adım Adım Çözüm

Factor the numerator of the expression in the target inequality.

\frac{x^2 - x}{|x - 1|} = \frac{x(x - 1)}{|x - 1|}

Factoring allows for analyzing the individual signs of the linear factors.

Analyze the denominator to simplify the inequality.

Since

|x - 1| > 0

for all

x \neq 1

, multiplying both sides of

\frac{x(x - 1)}{|x - 1|} > 0

|x - 1|

gives the equivalent inequality

x(x - 1) > 0

Multiplying an inequality by a strictly positive quantity preserves the direction of the inequality sign.

Solve the quadratic inequality

x(x - 1) > 0

The product

x(x - 1)

is positive when both factors have the same sign, yielding the solution set

x < 0

x > 1

A product of two real terms is positive when both terms are positive or both terms are negative.

Compare the rephrased target (

x < 0

x > 1

) with the proposed target (

x > 0

The range

0 < x < 1

makes

x > 0

true but makes

x(x - 1) > 0

false. Thus, the targets are not equivalent.

Two target questions are algebraically equivalent if and only if they yield identical truth values for all values in the domain.

Anahtar Kavram

Question Stem Simplification and Target Rephrasing

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