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When you have an X-ray it is done for a specific reason and it should not be feared; any potential risks should have been considered by your physician before the order to have an X-ray exam is written.
This dosen't mean you should not pay attention to what is done and how well you are protected by lead aprons or other shielding devices.
Some exams will not allow lead shielding due to the area of intrest being in a place that the lead would obscure. Seeing the body part is the reason for the exam and the lead can not block the path of the X-ray beam.
ALARA is the motto of radiologic technologists and other radiation professionals. It stands for As Low As Reasonably Achievable and means that the dose should be the minimum needed to do the job.