X-rays work because to x-rays your bones have a very different reflective quality compared to muscles, skin and tendons.

While you can see any damage to bones, or any sort of misalignment of bones (which is something that can happen if the tendons that fix the bone into place are damaged) you can't see the tendons themselves.

In cases where doctors do need to see tendons&ligaments they use Magnetic Resonance Imaging (MRI), but they tend to avoid that as MRI machines are super expensive to buy and use (although it's becoming more common, especially with hand, feet and knee injuries).