Let F \subset K \subset E be field extensions. Assume that F \neq K and that there is x \in E s.t E=F(x). Show that E is algebraic over K.

I have no idea how to even begin with this one. I'm sitting with my books trying to find some quality to use, but all are about finite extensions, something I suppose I can't assume here. As you know, if E=F(x) is finite over K then it is also algebraic over K, and I'm thinking about using the contrapositive, but that doesn't seem to solve the problem in its entirety.

Thank you for any advice.