What should i do about my cat?

ok well, i'm sitting with my russian blue cat, leonardo and i was petting him and then i ran my hand over his tail, it felt really strange and he cries when i tough it and he ripping the hair off of the spot to i think he broke his tail and i don't know what to do it obviously hurts him but every ones telling me even if i took him to the vets they wouldn't do anything for him because its just cartilage in his tail, im really worried im looking at it and its not bleeding or anything and im going to give him a bath and try to clean it or shouldn't i do that? im just worried and idk how much pain hes in because he never really liked it when i would touch his tail anyways so id really like some serious opinion's on what i should do. thanks.