It could be a handful of things. I'll list the most probable causes.
1.) The computer you inherited had a sound card, they took the sound card out, and never re-enabled the on-board audio in the system BIOS. To get into your system bios, re-start your computer and press F2 several times throughout the start-up. Find your on-board audio by pressing the arrow keys left or right until you see it and make sure it is not disabled, if it is diabled, put it to enabled.
2.) Almost all motherboards built within the past 10 years come with on-board audio and by the looks of the specs of your computer, it has on-board audio. It is possible it fried out on your motherboard, but highly unlikely. Do you have a little speaker in the bottom right of your computer? Double click it and make sure to drag down and select your audio device, if no speaker, then revert back to step 1 as it is most likely disabled.
3.) Click on start, click on control panel, click on sounds & audio devices, click on the hardware tab and check to make sure all the devices are working properly. Click on the other tabs (Audio, Sounds, etc) and make sure you put the drop down to your audio.
4.) If all of this fails you can run out to your local compUSA, best buy, staples and buy a creative sound card. If you want to get something that will blow you away get the creative x-fi (about 120 dollars). Creative soundblaster audigy 2 is a nice cheap card and is about 30 dollars and will do the trick. Also, make sure the speakers are plugged in to the correct spots :-).
Hope this helps.