Q: When driving my car and i am going about 35 or higher in speed my car sways back and forth in the back end.

This is a common concern caused by a broken sway bar or trailing arms in the rear of the vehicle. This is most definitely a safety issue as you may lose control of the vehicle in bad weather or at highway speeds. Consider hiring an experienced technician like one form YourMechanic who can come out and take a closer look at your vehicles suspension for a more personal diagnosis.