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2006 AMC 10B Problem 7

Problem 7 of 25EasierAlgebra

Which of the following is equivalent to x1−x−1x\sqrt{\dfrac{x}{1-\dfrac{x-1}{x}}} when x<0?x \lt 0?

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Solution

The denominator simplifies: 1−x−1x=x−(x−1)x=1x.1-\dfrac{x-1}{x}=\dfrac{x-(x-1)}{x}=\dfrac{1}{x}. So the expression is x1x=x2=∣x∣.\sqrt{\dfrac{x}{\frac{1}{x}}}=\sqrt{x^2}=|x|. Since x<0,x \lt 0, this equals −x.-x. Thus, the correct answer is A.
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