guerrero wrote:If -1 < x < 0, which of the following must be true?
I. x^3 < x^2
II. x^5 < 1 - x
III. x^4 < x^2
A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II and III
OAE
Constraint: -1 < x < 0.
Thus, x is a NEGATIVE FRACTION between -1 and 0.
I: x³ < x²
Since x≠0, x²>0.
Thus, we can safely divide both sides by x²:
x³/x² < x²/x²
x < 1.
Since x is a negative fraction between -1 and 0, it must be true that x<1.
Eliminate C, which does not include statement I.
III: x� < x²
Since x≠0, x²>0.
Thus, we can safely divide both sides by x²:
x�/x² < x²/x²
x² < 1.
Since x is a negative fraction between -1 and 0, it must be true that x² < 1.
Eliminate A and B, which do not include statement III.
II: x� < 1-x
x� + x < 1.
Since x<0, x� < 0.
Thus, x� + x = negative + negative = negative.
Thus, it must be true that x� + x < 1.
Eliminate D, which does not include statement II.
The correct answer is
E.
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