# 2006 AMC 10B Problems/Problem 7

## Problem

Which of the following 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 1 $\sqrt{\frac{x}{1-\frac{x-1}{x}}} = \sqrt{\frac{x}{\frac{x}{x}-\frac{x-1}{x}}} = \sqrt{\frac{x}{\frac{x-(x-1)}{x}}} = \sqrt{\frac{x}{\frac{1}{x}}} = \sqrt{x^2} = |x|$

Since $x<0,|x|= \boxed{\textbf{(A)}-x}$

## Solution 2

To confirm the answer, inputting a negative value into $x$ can help. For ease of computation, if $x=-3$, $\sqrt{\frac{-3}{1-\frac{4}{3}}}=\sqrt{\frac{-3}{\frac{-1}{3}}}=\sqrt{9}=3$. As no other option choice fits, $\boxed{\textbf{(A)}-x}$ is the correct solution. Note that if $x=-1$ was chosen, C would have fit as well. Make sure to avoid making this mistake.

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