Steering Wheel "shimmy" when braking
 

When traveling about 50 MPH or more, when I press on the brake pedal the steering wheel has a terrible "shimmy" (back and forth movement). I've changed the rotors, assuming they were warped, but that didn't help.
I've replaced one wheel bearing that was making noise, but I don't think that helped either. I've noticed now that the other wheel has some play both up and down and side to side.
I think the other wheel bearing needs to be replaced.
Times are tough, and I want to make sure I don't replace parts unnecessarily. I've heard of rack and pinion problems with these cars, but I don't know what the symptoms are for that problem.
Can someone please give me some simple things to look for to determine what's causing this "shudder" when I press the brakes at anything above 50MPH?

Thanks,

Have you had the alignment checked? Some places will check it for free

Thanks,
Nope that's all good.

The other hub needs to be replaced as well if there is that much slop and play in it.

Obviously, the other hub needs to be replaced. Would that cause this problem? I don't want to replace a hub that's going bad if that's not the problem. Money is tight and I'm just trying to replace the parts that are causing the problems in a logical order. If the rack is worn out, how would you tell?

Thanks,
It's tought being poor - but Obama's stimulus money hasn't found it's way into my pocket yet and I work in the construction industry. There's no construction going on right now.

Jack up the front of the car and check for play in all directions, up down, left right, back and forth, in and out. If you see or feel play, track down the source. Play will cause that area to jump about, and under braking it can get worse. My guess is the wheel has play in the steering and braking is causing it to flutter back and forth.