The answer to your question is yes, it can be done, but for the average RVer it is not practical. To put it simply RV Air Conditioners have high wattage demands at start up and moderate to high wattage demands while running. The average house battery systems in RVs do not have enough DC battery power (not enough batteries) to support these demands when using an inverter.
So you would have to add additional house batteries. How many batteries you have to add depends on how many watts your AC uses on startup and while running and of course additional batteries means additional weight to the RV and also figuring out where to put the additional batteries.
You would also need to beef up the alternator on your Motorhome to meet the additional charging needs of the house batteries. You would also need a Pure Sine Wave Inverter (which are more expensive then the commonly used Modified Sine Wave Inverter) with a high enough wattage rating to meet the start up power demands of your air conditioner. When an AC unit starts up it can require up to 3 times more power to get it started then it uses when it is running.
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