This can happen on some laptops. Lets begin by trying:
sudo apt-get install pulseaudio pavucontrol
This typically has a 50% chance of working.
Now what I am going to do is copy a comment by Mark Greaves that he has left on the peppermint official forums:
is the sound card listed by:
If not, can you run:
You'll then find a file in your Home folder called dmesg.txt.
Post it's contents on paterbin HERE
and provide a link to it in your next posting