Turn the computer on and point a flashlight on the screen.
If you can see stuff in the background is most likely an inverter issue.
If you cannot, it is likely to be an issue with the LVDS cable.
before you proceed, just make sure you can get a video output on an external monitor so you can discart any kind of gpu failure.
Furthermore, Lenovo makes their service manuals available, [http://tim.id.au/laptops/lenovo/thinkpad%20sl400%20sl400c%20sl500%20sl500c.pdf|here] is yours.