In Data Sufficiency, simplifying the question target "Is x^2 < x?" yields the algebraicall… — GMAT

In Data Sufficiency, simplifying the question target "Is $x^2 < x$ ?" yields the algebraically equivalent question target "Is $0 < x < 1$ ?" for all real numbers $x$ .

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Cevap

The statement is true because solving the inequality

x^2 < x

yields the exact range

0 < x < 1

Rephrasing the question stem target

x^2 < x

algebraically leads directly to

x(x - 1) < 0

, which holds true if and only if

x

is strictly between

0

and

1

. Therefore, the simplified target 'Is

0 < x < 1

?' is completely equivalent.

Adım Adım Çözüm

Rearrange the inequality to set one side to zero.

x^2 - x < 0

Standard algebraic procedure for solving non-linear inequalities requires comparing a factored expression to zero.

Factor the quadratic expression.

x(x - 1) < 0

Factoring isolates the roots (

x = 0

and

x = 1

) that define the boundary intervals on the real number line.

Determine the interval where the product of the factors is negative.

The product is negative between the roots, which corresponds to

0 < x < 1

When

x < 0

, both factors are negative (product is positive). When

x > 1

, both factors are positive (product is positive). Only when

0 < x < 1

x

positive and

(x - 1)

negative.

Anahtar Kavram

Simplifying Data Sufficiency question stems by finding equivalent inequality ranges reduces complex targets to direct boundary checks.

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