Checking Ochem Reaction Validity: 1-Methyl-1-Cyclohexanol with HBr and EtO-/EtOH

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tandoorichicken
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I did this problem a little differently than my professor, so I need someone to check if it is valid.

Let's start with 1-methyl-1-cyclohexanol. If I react this with HBr, I should get 1-bromo-1-methylcyclohexane. And if I then react this with EtO-/EtOH, I should get 1-methyl-1-cyclohexene, right?

My professor wrote a reaction with sulfuric acid to give a carbocation that reacted with water to give the same product.
 
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You are going to need to consider whether the carbocation site would transfer itself in this case, also, remember that the elimination mechanism takes place in trans.