2012 AMC 12B Problem 6

In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?

\textbf{(A)}\ \text{Her estimate is larger than }x-y

\textbf{(B)}\ \text{Her estimate is smaller than }x-y

\textbf{(C)}\ \text{Her estimate equals }x-y

\textbf{(D)}\ \text{Her estimate equals }y - x

\textbf{(E)}\ \text{Her estimate is 0}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.