A home inspector may not have the professional training of a pest control professional and thus may not have performed a "termite inspection" as would be done by a pest control professional. If you paid for a professional pest control "termite inspection" then you should read the agreement concerning the inspection and see what it covers. It is entirely possible for a home to have no visual indication of a termite problem one month and then 3 months later have a problem. That is why a termite bond around North Carolina is a must.
I would have a pest control company (not the same one used in the inspection, if one was used) to come out and look at the infestation. Have them give you an opinion of when they think the infestation started or how long it had been there. If the infestation appears to have been there prior to the inspection, then I would show the previous inspection company the data and ask them what they plan to do.
Unfortunately this is not a cut and dry situation. There are a number of things that need to be determined before you can look to the inspection company for a remedy.
To be fair, it is very difficult to see termite infestations under slabs and behind walls. Unless there is visual evidence most inspections will not find active termites.
Real Estate Advisor/Broker
Then I would recommend contacting the home inspector and politely telling them about the situation and informing them that you are not pleased and see if there is anything they will do to help you out.
I don't know what the agreement or terms were with your home inspector when they performed the inspection, but if they don't agree to help you then you may need to contact an attorney to see if you have any recourse. Did the home inspector specifically say that their inspection would include a check for termites?
In my experience from attending home inspections, I think that a termite check is usually conducted inside the basement by probing at the main support beams with a metal rod around the perimeter of the home to check for soft rotting wood. Since the house is on a slab the only way to check would have been to remove the siding from the house. That may be why they did a minimal inspection.
I had a home inspection turn up termites once and it killed the deal, but I'm glad the inspector caught it for my buyers as they were able to back out. I hope this works out for you. It is not a fun situation to be in.