I've got an old computer which still has some life in it. The motherboard has an AGP 1x/2x slot, and I have an unused AGP 2x/4x card laying around. Could I use this card with this motherbaord? My reasoning is that they both have the "2x" in common. So basically, would this pairing work?
Yes it will work.
The only problem I have ever seen with AGP is some of the 8X cards will not physically fit in some of the older AGP slots.