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 True

D. 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