Carcass wrote:
If \(\frac{1}{x} <x <0\) , which one of the following must be true?
A. \(1<x^2\)
B. \(x^2<x\)
C. \(-1<x^3<0\)
D. \(\frac{1}{x} >-1\)
E. \(x^3<x
\)
Kudos for the right answer and explanation
We have: \(x < 0\)
Also, \(x > 1/x\)
Thus, x must be a negative fraction between 0 and -1
For example: \(x = -1/2 => 1/x = -2 => -1/2 > -2\)
Thus: \(-1 < x < 0\)
Working with the options:
A. Since \(x\) is a negative fraction between 0 and -1, \(x^2\) must be a positive fraction between 0 and 1 => \(1<x^2\) is False
B. Since \(x\) is negative, and \(x^2\) is positive, \(x^2\) must be greater than x => \(x^2<x\) is False
C. Since \(x\) is a fraction between 0 and -1, \(x^3\) will be a negative fraction also lying between 0 and -1 => \(-1<x^3<0\) is
TrueD. Since \(x\) lies between 0 and -1, \(1/x\) must be less than -1 => \(\frac{1}{x} >-1\) is False
E. Since \(x\) is a fraction between 0 and -1, \(x^3\) will be a negative fraction but having a smaller magnitude, i.e. \(x^3\) is greater than \(x\) => \(x^3<x\) is False
Answer C