Edit. Sorry, that doesn't work. Just because x is not congruent to 1 it doesn't follow that a is not congruent to 1. But maybe you can see how to fix that by choosing x more carefully?
Second edit. I think I see how to fix it. The multiplicative group of nonzero residue classes mod p is cyclic, so you could take x to be a generator. Then will not be equal to 1 mod p.