If the system is still in warantee call Dell before messing with it. They are good about warantee support. If it's out of warantee:
Sounds like a video card problem or maybe just a bad video cable connection. Look at the back of the case. There are metal strips from side to side where expansion cards plug in. If the monitor cable is plugged into one of these then you have a seperate video card. If the monitor plugs in near where the LAN, USB, printer ports etc are then you have built in video.
If it is a video card you need to open the case and make sure the video card is plugged all the way into it's socket. Push down on the top edge of the card. Also make sure the screw that holds the video card to the back of the case is there and tight. Plug the monitor back in and try it again. If it stays on the video card was just loose. It happens sometimes if the cable gets yanked on.
If the video is on the MB there is not much you can check. You can borrow a video card from someone and try it to see if an add in card works ok. be advised this might mess up your on board video drivers so be prepared to have to reload it's driver after trying the add in card.
Download cpuz.exe (available at
www.majorgeeks.com) and run it. It will read your system and provide details on your memory, processor, motherboard chipset, etc. You can find out about your video by opening start-control panel-system-device manager and looking under the display adapters line.
I found a link on the optiplex. It's about 3 years old model but with a 3 year warantee. The video is built into the MB and some of the optilex models don't look like they will accept a plug in video card.