What would cause the daytime running lights and the drivers side hi-beam to not work on a 98 Chevy Cavalier?

Check the wiring and socket to the high beam light. Mine was bad on a 96 and after replacing it mine worked.

did you ever figure out what was the matter with the car....? curious....as i have a 2000 that the daytimes go out on ...and do not have any high beams.....thanks

My did the same, you have to replace the regular light bulb on the drivers side, it should work after that. Is your service light (on your dashboard) on?