Inventively tells you HOW someone did something: in a fresh, interesting way, not the usual way. She solved the problem inventively means she used a new method. The meaning is the same family as inventive, but this form does not describe the person. It describes the action. You can cook inventively, dress inventively, or use a small space inventively. It is a slightly careful word; in very simple talk people often just say in a clever way or in a new way.