Bunuel wrote:

If \(x = −\frac{2}{3}\), which of the following inequalities properly lists its terms in ascending order?

A. \(\frac{1}{x}<x<x^2<x^3\)

B. \(x<x^3<\frac{1}{x}<x^2\)

C. \(x^3<x^2<x<\frac{1}{x}\)

D. \(x<x^2<x^3<\frac{1}{x}\)

E. \(\frac{1}{x}<x<x^3<x^2\)

E

Courtesy: Ron

"if you ever see a problem like this -- on which you're comparing different powers of the same variable -- then, immediately, you should think about the following number properties:

* signs

* ""fractions"" (between 0 and 1, or between -1 and 0) vs. ""non-fractions"" (greater than 1 or less than -1)

for each of these ranges of numbers -- less than -1; between -1 and 0; between 0 and 1; greater than 1 -- the powers have different properties, and so the ordering of the powers is different.

it's rather pointless (and possibly confusing) to memorize what happens to the powers. instead, just throw in a few numbers of each type and watch what they do.

if the inequalities are not strict inequalities (i.e., if they are the < or > type), then you should also mess around with the numbers 0, 1, and -1."

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आत्मनॊ मोक्षार्थम् जगद्धिताय च

Resource: GMATPrep RCs With Solution