I dont know the price. But you can actually do braces without x-ray if the dentist is good enough.
Usually x-ray is only necessary if your teeth are really badly arranged, and need a lot of planning.

X-ray record is good to keep if you are changing dentist, so the future dentist to work on you have an overview of how your teeth used to be.
Other than documentation purposes, you can save money if you dont do x-ray.

My dental record only used picture documentation.