Nice One : Is x negative?

If 0 < x < 1, then x^3 < x^2 < x.

If x > 1, then x < x^2 < x^3.

So when x is positive, x^2 is never the largest of the three expressions x, x^2 and x^3. Statement 1 tells us that x^3 is not the largest of the three expressions, and Statement 2 tells us x is not the largest of the three expressions. So x^2 must be the largest, and thus x cannot be positive. So the answer is C.

Note that neither statement is sufficient alone, since while x can certainly be negative using either statement, using statement 1, x could also be positive (x could be 1/2, say), and using Statement 2 x could be positive (x could be 2, say).

hey, i wrote that problem!

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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.

Not to ditto everything that Ron is saying today...

I often tell my students that whenever they see a problem with exponents and inequalities, they should draw this number line, to make sure they don't forget positives/negatives or fractions/non-fractions:



If you quickly test each of these 7 numbers, you'll have considered all of the types of numbers that behave differently with inequalities and exponents.

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