Computer thinks it has no sound card unless headphones plugged in

Unless I have speakers or headphones plugged in, my computer gives me the red "X" over my sound icon in my system tray, and anything that involves sound won't work. This is annoying because I don't always want to listen to sound at work, but I still need to be able to play things that have sound while I listen to music from another source.

I guess I could just find a wire to plug into the headphone jack (yup, that works) but that seems silly. What gives? I've never had this problem on any other computer.

That's normal behavior from what I've seen. If not output devices are detected, the sound is disabled. I'm not sure of the part you are stuck on, but if there's no output device connected, then you obviously aren't going to hear any sounds at all. Your computer likely doesn't have an internal speaker, either. If you want to hear sounds, plug in speakers or another output device.

Never seen one do that. I don't want to HEAR the sounds, I just want the computer to be able to play things that involve sound. For example, at work I edit video. There is sound on the sound tracks, but I don't need to hear them. But without thinking it has a sound card, the video won't even play for me to edit.

You can always plug in speakers and leave them off. You could also check the driver settings to see if this behavior can be changed, but I don't recall seeing that option on any OEM system I've been working on recently. My company uses Dell, and that's how it works for them.