Will the system work on a laptop with a bad graphics card? Windows 7 Home Basic operates normally if you do not install the video card driver. The integrated card does not work and the second is fine.
The system should detect the second graphics card, though you may need to disable the first (depending on how it's detected and why it's broken). Not sure which cards you have, but you might need to install a proprietary driver to improve performance after the original install.
You can test the OS without installing by following the install directions, booting from the USB or CD, and following "Try elementary" rather than "Install". It will boot into the OS for you to test without committing anything to hard drive.