First thing to do is leave that hair dryer at home and buy one that is half the wattage. Thats what I told my wife to do and she uses hers every day. It has 2 settings and she dries her hair on the lower setting(900W). Works fine. Our micro is around 1000W and we use it not for cooking but to heat up things, popcorn etc. We had a 2000W xantrex Freedom PSW inverter.
4 batteries, 330W solar worked fine. Used the inverter for small things, magic bullet, coffee grinding, toaster, Mac book charging also as previously stated, micro and hair dryer plus camera, iPad, phone battery charging. on/off switch was easy to reach so when the job was done, the inverter was turned off.
Dave