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2010 AMC 12A Problem 4

Problem 4 of 25EasierAlgebra

If x<0,x\lt0, then which of the following must be positive?

Answer choices

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Solution

Choice (D) is −x−1=−1x.-x^{-1}=-\dfrac1x. When x<0,x\lt0, 1x<0,\dfrac1x\lt0, so −1x>0.-\dfrac1x\gt0. Testing x=−1x=-1 shows the other choices need not be positive: x∣x∣=−1,\dfrac{x}{|x|}=-1, −x2=−1,-x^2=-1, −2x=−12,-2^x=-\tfrac12, and x3=−1.\sqrt[3]{x}=-1. Thus, D is the correct answer.

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Concepts: inequality · exponent · absolute value

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.