2006 AMC 10B Problems/Problem 7

Problem

Which of the folowing is equivalent to $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ when $x < 0$

$\mathrm{(A) \ } -x\qquad \mathrm{(B) \ } x\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } \sqrt{\frac{x}{2}}\qquad \mathrm{(E) \ } x\sqrt{-1}$

Solution

$\sqrt{\frac{x}{1-\frac{x-1}{x}}} = \sqrt{\frac{x}{\frac{x}{x}-\frac{x-1}{x}}} = \sqrt{\frac{x}{\frac{x-(x-1)}{x}}} = \sqrt{\frac{x}{\frac{1}{x}}} = \sqrt{x^2} = |x|$

Since $x<0$

$|x|= -x \Rightarrow A$

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2006 AMC 10B Problems/Problem 7

Problem

Solution

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