Is it possible to mine both LTC and BTC in the same rig? I guess the root question would be is it possible to prevent bitcoin miners from using the graphics card and only utilizing the ASIC? Or would I have to buy a separate rig altogether?
ASICs are standalone machines, so that's not an issue. A lot of people build dedicated mining rigs with the cheapest possible hardware apart from the expensive GPU they use to mine.
You wouldn't have to buy a separate rig, but the rig you use would be somewhat tied down by mining if you plan to do it 24/7. To be honest, you should research mining extensively and be aware of the risks. It's difficult to make money day by day because you have to make more in crypto-value than you pay for electricity, let alone re-couping the cost of your mining hardware, let alone making a profit on the venture.
But it can be fun and rewarding, so keep reading!
This is the mining board:
https://bitcointalk.org/index.php?board=14.0