How do I attach a small accent rug to my carpeted house

You need the carpet underlay which should help to stop the sliding.

If you put a rubber backing on it that should help. There are rubber type mats sold for lining kitchen cupboards I believe. Just lay it down and lay the accent rug on top and see how that works. If all doesn't help try stitching the rubber to the back of the accent rug.

rug grippers -- kind of soft rubbery 'foam' stuff -- like little pillows -- in the rug/carpet section of 'walmart' ...

For the wood floors, I just put hot glue in strips on the back of the rug. It doesn't move anymore unless I move it. It would probably work on the carpet too. You could also hot glue those non skid pad things right to the back of the rug

You wouldn’t want to put a foam or adhesive stabilizer under a rug for hardwood floors because they might stick and leave a residue that is hard to get up, but might also stain the wood or sealer. But the same backings might work on carpet. On the other hand, throw rugs come with no backing, an applied backing, and thick foam backings. If the one rug that slides now doesn’t do any better with one or two of the solutions, like backing tape or a foam waffle stabilizer, and if you can’t weight each corner with furniture, use the rug in another place in the house. There is always use as a wall hanging, giant island runner, or an upholstery protector. Best wishes ☺️

there are runners that you place under the rug that help keep the rug in place you can find them at any of your local hardware store. or you can just use good old duct tape roll a little under and put under each corner.

Lol...sorry, just remembering my solution to the traveling rug. I nailed it to the floor under the carpet