2006 AMC 10B Problem 7 — Solution | LIVE by Po-Shen Loh

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2006 AMC 10B Problem 7

Below is the professionally curated solution for Problem 7 of the 2006 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2006 AMC 10B solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:radical absolute value algebraic manipulation

Difficulty rating: 1240

7.

Which of the following is equivalent to

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

when $x \lt 0?$

$-x$

$x$

$1$

$\sqrt{\dfrac{x}{2}}$

$x\sqrt{-1}$

Solution:

The denominator simplifies: $1-\dfrac{x-1}{x}=\dfrac{x-(x-1)}{x}=\dfrac{1}{x}.$

So the expression is $\sqrt{\dfrac{x}{1/x}}=\sqrt{x^2}=|x|.$ Since $x \lt 0,$ this equals $-x.$

Thus, the correct answer is A.

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Problem 7 in Other Years

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