The inequality \(|z - 3 - i| \leq 3\) represents a circle centered at \((3, 1)\) with radius 3 on the Argand diagram.

The inequality \(|z| \geq |z - 4i|\) represents the region above the line \(y = 2\) (since \(|z|\) is the distance from the origin and \(|z - 4i|\) is the distance from the point \((0, 4)\), the line \(y = 2\) is the perpendicular bisector of the segment joining these points).

Therefore, the solution is the region inside the circle centered at \((3, 1)\) with radius 3, and above the line \(y = 2\).