RALEIGH, N.C.—Elon Musk made a fortune disrupting the status quo in online shopping and renewable energy. Now he's up against his toughest challenge yet: local car dealers.

Mr. Musk, the billionaire behind PayPal and now Tesla Motors Inc., wants to sell his $70,000 Tesla electric luxury vehicles directly to consumers, bypassing franchised automobile dealers. Dealers are flexing their considerable muscle in states including Texas and Virginia to stop him.

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