2006 AMC 10B Problem 7

Problem 7 of 25EasierAlgebra

Which of the following is equivalent to

\sqrt{\dfrac{x}{1-\dfrac{x-1}{x}}}

when

x \lt 0?

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Solution

The denominator simplifies:

1-\dfrac{x-1}{x}=\dfrac{x-(x-1)}{x}=\dfrac{1}{x}.

So the expression is

\sqrt{\dfrac{x}{\frac{1}{x}}}=\sqrt{x^2}=|x|.

Since

x \lt 0,

this equals

-x.

Thus, the correct answer is A.

Concepts: radical · absolute value · algebraic manipulation

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