AMC 12 2010a Problem 6

A $\textit{palindrome}$, such as 83438, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?

\textbf{(A)}\ 20 \textbf{(B)}\ 21 \textbf{(C)}\ 22 \textbf{(D)}\ 23 \textbf{(E)}\ 24