In all pprobibilty, you don't have to make a profit to be considered a business, just having a business name would make you a business. Cashing the check isn't the problem, the problem is if the client uses it as a business deduction.
The advice to take a business course is good. So is the advice not to discuss this on a public forum ... you really don't know who is monitoring it. OPSEC (Operations Security). Remember, loose lips sink ships.
It may be beneficial for you to be a small business. There are lots of deductions that you can take, BUT it can be a pain to do all the record keeping and red tape of making it happen.