DX body. 10-24mm f/3.5-4.5 & 24-120mm f/4 good combo?

I recently sold my 18-200mm and ordered the new 24-120mm f/4. I also currenly own the 10-24mm f/3.5-4.5.

My primary reason for selling the 18-200mm was that I was not been satisfied with the sharpness of the images that it produces. I have read great things about the sharpness of the 24-120mm, and prefer buying full frame lenses so I can upgrade to an FX body in the future without losing my investment in glass.

Now that I'm waiting for the 24-120 to arrive, I'm having second thoughts about having to swap lenses with the 10-24mm at the 24mm focal length. This seems like a frequently used focal range on a DX body.

Is the 10-24mm & 24-120mm f/4 a good combo for a DX body?