Leveling a concrete floor

The concrete slab in my family room settled resulting in a crack down the middle and causing each end to sink about 2 inches. I want to make it level so I can lay tiles. I want to make the floor close to level then cover it with leveling compound.
My question is: What is the best way to make it "close to level" so I don't have to use so much leveling compound (which is expensive and not recommended for thick application)?
Does the rule that concrete has to be 2 inches think apply even if its going over a slab that is already 2+ inches thick? Can I just pour cement in the deep areas and maybe throw in some aggregate so I don't use as much cement? I'm even thinking of burying some extra "Wonderboard" in the deep areas and cementing it to the floor. Does it really matter how I fill in the deep areas before I pour leveling compound?
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Sure the ends sunk, and the middle didn't heave higher? Is this a converted garage/patio, or was the room built using a proper perimeter foundation, and the slab poured inside? Does the high spot move around summer vs. winter, or dry season vs. wet?
Sorry to be the bearer of bad tidings, but any attempt to kludge the existing slab will be doomed to failure, unless you fix what caused it to move in the first place. It will keep moving. At a minimum, you need to get a mud-jacking company and/or engineer in there, to figure out what happened. Can't see from here, but from your description I suspect the slab is too thin and/or wasn't reinforced correctly and/or the site prep/drainage was inadequate.
Personally, if I was gonna sink a lot of money into a fancy tile job, I'd be inclined to jackhammer out existing slab, and pour a good one. Unlike tile, concrete is cheap. The labor to hammer it out and pour a proper slab won't be much more than the solution you propose, and the end result will be MUCH nicer.
aem sends...
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I would not tile unless you are sure it wont move any more, it likely will move as something was poorly done.
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I bought this house 4 months ago. It was built in 1980. The family room is below grade by two feet. A perimeter foundation was poured and then a slab poured. I had an inspection by an structural engineer. The foundation is stable. The slab is what shifted and yes the middle of the room is higher than the ends. I was planning to use the middle of the room as my starting point for level. I haven't been in the house long enough to know if it shifts during the seasons.
The structural engineer said it looks like it settled shortly after the house was built. He theory is that the fill used under the slab was not compacted enough.
Do you think I could level it with cement and then a leveling agent and put carpet instead of tile on the floor? I really don't want to go to all the work of jack-hammering the old slab and removing it.

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I think carpet or linoleum is probably a wise choice.
Dave

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Is the slab entirely floating, or do the outside edges rest on the footer or a ledge of the foundation like they should? That is why I think the middle probably came up. But in any case, if your engineer thinks things are stable now, yeah, you could probably pour another layer of concrete over it, assuming there won't be any trouble with doors and stuff. Don't go higher than the the sills of the wood walls (if they go below grade), better to keep it a few inches below. (concrete should never be higher than wood.) I'd do a thick enough layer to put 4-inch mesh in it. I'd also go with cheap (non-glued) carpet, maybe carpet tiles, for at least a full year, and keep an eye on it, before you consider putting down an expensive floor. Watch for damp, new cracks, more heaving, etc.
aem sends...
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Try a google search on Gypcrete I think it would be perfect for what you want to do. My friend just used it in three rooms over wooden floors to level out the floor and he also installed radiant in floor heat in it. It goes on very thin and is almost self leveling.

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level
don't
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cement?
and
When you say "tile" are you talking ceramic? If so, why not put a mud bed in over a membrane? A floating floor! Do a search on TCA F-111 for more info.
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