Solving equations over unknown Domains

I am trying to solve for what values of some variables make some expressions true; not exactly equations.

For a toy example, I am trying to solve for all values of $x$ that make $4x$ an Integer.

I have tried Solve[4x [Element] Integers, x] and similarly Solve[Element[4x,Integers]==True,x], but they don’t seem to work.

For simple expressions such as above, this is not hard to do at all by hand or in one’s head, however when solving for multiple variables and multiple nonlinear expressions, it becomes tricky.