I have done a few of those, it's pretty easy.
You dig a hole on one side, about 8" deep. The hole only needs to be big enough for a floor jack. A small floor jack will work just fine.
Then, undermine the concrete enough to put the end of the jack under the slab.
If the ground isn't firm, use a piece of wood or heavy gauge steel, under the front of the jack. Otherwise the wheels will just sink into the ground, and you can end up with a bent jack.
Next, jack up the side of the slab, and insert your fill. 304 lime stone sand gravel mix, some call it "crusher run", is ideal. But, just limestone sand, or dry concrete mix will work. play sand, or mason sand, will attract ants, so it's not ideal to use here if it's near you house.
You can also put bricks, or treated wood in areas, if it's really sunk, to help stabilize it while things settle.
It may take a couple of tries to get it just where you want it. And, you may have to do both sides of some pieces. But, without a lot of effort, you can fix this really nicely. Even the grass can be put back if you remove it carefully.
The only thing you need to be sure of, is that the control joints are cleanly broke all the way through. If not, when you jack it, it will bind up, and it won't move. If that happens, you need to let it back down, and saw cut the ends all the way through cleanly. If you don't, you may end up with them breaking off, and it will be all chipped up.
I once leveled a couple of sections of regular sidewalk for someone, by just putting pieces of treated 2x4's under them, and it was fine decades later. Grass, dirt, and sediment will eventually get under there and fill any voids.
