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2001 AMC 10 Problem 2

Problem 2 of 25EasierAlgebra

A number xx is 22 more than the product of its reciprocal and its additive inverse. In which interval does the number lie?

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Solution

The reciprocal of xx is 1x\dfrac1x and its additive inverse is −x.-x. Their product is 1x⋅(−x)=−1.\dfrac1x\cdot(-x)=-1. So x=2+(−1)=1,x=2+(-1)=1, which lies in the interval 0<x≤2.0\lt x\le2. Thus, the correct answer is C.

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Concepts: fraction · algebraic manipulation

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