About three weeks ago part of my tooth broke off. I do not have insurance so I could not go to a dentist. It was very painful for two weeks then this past week it did not hurt at all. Then two days ago I woke up and my cheek was swolen(the side the broken tooth is on). Its not painful but I do not know what to do. Could this be a serious problem?

Yes you need to get to the dentist it sounds like you have an abscess.

Get it pulled out it's cheaper. I have the same problem and can't afford to get a root canal and crown, it's cheaper to get it pulled out. Should not run you more than $ 150.00

Definitely sounds like you have an abscess. You aren't having any pain right now as the abscess must be draining. You are probably swallowing the pus that is draining out of it. This pus can make you quite ill. See a dentist as soon as you can for an antibiotic and tooth removal if you financially can't save it.

All the best!