I recently went through this situation-about two weeks ago. Not only was I installing carpet over tile, but I was installing tile over tile. I went to Home Depot and looked in the area where they sell underlayment for floors and purchased a bagged powder that you mix with water to get underlayment (7.5lb bag is $14 if I recall correctly.) This can also be used for "transition areas" like where there is tile and where there is not. At first I did not use it under the carpet, the carpet was walked on and settled and since I knew where to look, I could see a line where the tile started. I pulled up the carpet and put in the underlayment and feathered the edges and it looks great now. Good luck.