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I do suggest an x-ray. A doctor will be able to tell you if there is a fracture. They will also be able to tell from the x-ray if the fracture is displaced. Sometmes some manipulation and casting needs to be done.
It could also be a bad sprain, but only an x-ray will be able to tell.
Explain to your doctor that you have no insurance. I'm sure your regular doctor will be able to come up with a payment plan you both can agree on.