You can always fight every type of charge. It is your right. A moveable offense like a seatbelt violation will go onto your record if you have a conviction. If you want to plea to the case if you take defense driving or plea to a deferred you can get the case taken care of without having to have it count as a point on your license. If you want to fight it you can contact an attorney to help you fight the case. It sounds like you have some good issues with the case that can result in a dismissal. It just depends if you want to hire an attorney vs going in and taking defensive driving or a deferred.
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Depends if it will go on your record. You have to see how the City of Houston plead the ticket. Some seat belts tickets do not go on your driving record while others do. I see it all the time people waste their money and their defense driving on a seat belt ticket when it usually doesn't go on your record. Yes you can fight your ticket, that is your consitutional right in the State of Texas to fight it.