Square A and Square B are both $2009$ by $2009$ squares.  Square A has both its length and width increased by an amount $x$, while Square B has its length and width decreased by the same amount $x$.  What is the minimum value of $x$ such that the difference in area between the two new squares is at least as great as the area of a $2009$ by $2009$ square?
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Square A and Square B are both $2009$ by $2009$ squares.  Square A has both its length and width increased by an amount $x$, while Square B has its length and width decreased by the same amount $x$.  What is the minimum value of $x$ such that the difference in area between the two new squares is at least as great as the area of a $2009$ by $2009$ square?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{2009}{4}
Square A and Square B are both $2009$ by $2009$ squares.  Square A has both its length and width increased by an amount $x$, while Square B has its length and width decreased by the same amount $x$.  What is the minimum value of $x$ such that the difference in area between the two new squares is at least as great as the area of a $2009$ by $2009$ square?
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