1978 AHSME Problems/Problem 9

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Problem 9

If $x<0$, then $\left|x-\sqrt{(x-1)^2}\right|$ equals

$\textbf{(A) }1\qquad \textbf{(B) }1-2x\qquad \textbf{(C) }-2x-1\qquad \textbf{(D) }1+2x\qquad  \textbf{(E) }2x-1$


Solution

We have $\sqrt{x^2} = |x|$, so we rewrite the expression as follows. \[|x - \sqrt{(x-1)^2}| = |x - |x-1||\] We know that $x < 0$, so $x-1 < 0$. Thus, we can rewrite $|x-1|$ as $1-x$. So \[|x - |x-1|| = |x - (1-x)| = |2x - 1|\]. Since $x< 0, 2x-1 < 0$. Thus, we can write this as \[|2x - 1| = 1- 2x\] $\boxed{B}$


~JustinLee2017


See Also

1978 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
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