Step 1: Locate the available beta-hydrogens.
In 1-bromo-1-methylcyclohexane, the bromine and a methyl group both sit on carbon 1 of the ring. For elimination we need a hydrogen on a carbon next door, and there are two different sets available, the hydrogens on the methyl group itself, and the ring hydrogens on carbons 2 and 6.
Step 2: Write out the two possible alkenes.
Removing a hydrogen from the methyl group would put the double bond outside the ring, giving methylenecyclohexane. Removing a hydrogen from carbon 2 or 6 instead keeps the double bond inside the ring, giving 1-methylcyclohexene.
Step 3: Decide which one wins using Zaitsev's rule.
Zaitsev's rule favours the alkene with more alkyl substitution on its double bond carbons, since a more substituted double bond is more stable. 1-methylcyclohexene has three alkyl groups on its double bond carbons, trisubstituted, while methylenecyclohexane has only two, disubstituted.
Step 4: Conclude.
Being the more substituted, more stable alkene, 1-methylcyclohexene forms as the major product.
\[ \boxed{\text{1-methylcyclohexene}} \]