If -1 < x < 1 and x ≠0, which of the following inequalities must be true?
I. x³ < x
II. x² < |x|
III. x� - x� > x³ - x²
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III
Since -1 < x < 1 and x ≠0, x must a NEGATIVE OR POSITIVE FRACTION.
Statement I: x³ < x
If x = -1/2, then x³ = -1/8.
In this case, x³ > x.
Since it does not have to be true that x³ < x, eliminate A and E.
Statement II: x² < |x|
Since x is nonzero, x² > 0 and |x| > 0.
Since both sides of the inequality are positive, we can square the inequality:
(x²)² < (|x|)²
x� < x².
Since x² > 0, we can divide both sides by x²:
x�/x² < x²/x²
x² < 1.
Since the square of a negative or positive fraction must be less than 1, statement II must be true.
Eliminate C.
Statement III: x� - x� < x² - x³
Since x is nonzero, we can divide by x², which must be a positive value:
(x� - x�)/x² < (x² - x³)/x²
x² - x³ < 1-x
x²(1-x) < 1-x
Since x is a negative or positive fraction, we can divide by 1-x, which also must be a positive value:
x²(1-x)/(1-x) < (1-x)/(1-x)
x² < 1.
Since the square of a negative or positive fraction must be less than 1, statement III must be true.
Eliminate B.
The correct answer is
D.
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