Yes, but many times people create separate entities such as a corporation or a limited liability company to operate each business to diversity assets and/or businesses into different entities for asset protection purposes. You know what happens if you put all of your eggs in one basket and drop your basket - you lose everything.
I could form an LLC and have it own and operate two successful businesses. Business 1 is a popular bar worth $100,000. Business 2 is a car wash worth $100,000. A customer of the bar is over served and causes a car accident that kills his passenger. The family of the passenger sues my LLC and wins a big judgment that exceeds the LLC's insurance coverage. Result: I lose $200,000 of value.
If I had created two separate LLCs and operated each business in its own LLC, then the plaintiff in the lawsuit would have obtained only the equity in the LLC that owned the bar business. My car wash business would not be affected. Result: I lose $100,000 instead of $200,000.
If you start two businesses and neither has much value, you might as well operate them in the same entity. If over time, however, they appreciate significantly in value, at some point you should split them into separate entities to diversity to get better asset protection.
My response to your question(s) is not legal advice for your specific situation. My response discusses general legal concepts. The fact I responded to your question(s) does not create an attorney client relationship between you and me. Because I am licensed to practice law only in Arizona, my statements are general statements about Arizona LLC law rather than the law of any other state. The facts of your situation and the applicable federal and state law must be considered before taking action. I recommend that you contact an experienced attorney who is licensed to practice law in the jurisdiction that will govern your specific transaction and who has experience with the specific areas of law applicable to your situation.
Please see my answer to this same question at http://www.avvo.com/legal-answers/how-do-i-organize-separate-business-activities-und-296053.html#answer_392999
Can you - yes. Should you - no. You should set up a separate entity for each business to segregate the liability associated with each activity.
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DISCLAIMER: THE INFORMATION PRESENTED HERE IS GENERAL IN NATURE AND IS NOT INTENDED, NOR SHOULD IT BE CONSTRUED, AS LEGAL ADVICE. THIS POSTING DOES NOT CREATE ANY ATTORNEY-CLIENT RELATIONSHIP BETWEEN US. FOR SPECIFIC ADVICE ABOUT YOUR PARTICULAR SITUATION, CONSULT A QUALIFIED ATTORNEY.