If you have more than a finite number of points, yes, that is an infinite number of solutions. Any area will be other than finite.
Suppose we have a system of inequalities that we graph and the solution is a bounded region. Wouldn't that still have an infinite number of solutions? (For instance, if we have a triangle at the origin, there technically infinitely many points in there (i.e. (0.00001,y), (0.000000001, y2)) etc?