A question about uncountable, dense sets in R

A question about uncountable, dense sets in R

If you have an uncountable subset of $\mathbb{R}$, which is dense in
$\mathbb{R}$, is its complement countable?
I'm trying to see if you can take the set of irrationals, remove a
countable amount, and still have a dense subset in $\mathbb{R}$.If this is
the case, this sets complement would still be countable.
My goal is to determine if you can have an uncountable subset of
$\mathbb{R}$, which is dense in $\mathbb{R}$, yet its complement is
uncountable.
If somebody could just let me know if this is possible or not I would
appreciate it. Thanks.