After typing in my password, the startup loading screen goes 1/3 of the way through and then shuts off.

I can't login to my user account. I can type in my password, but the loading screen only makes it 1/3 of the way through before my computer shuts off. I'm running OS X Yosemite 10.10 on an 11-inch 2010 MacBook Air.
Not sure if it's relevant, but I seem to be having some graphical issues as well. When I scroll, text is sometimes jittery, and the corners of windows are black squares.
I've tried going to Recovery Mode and reinstalling OS X, but it tells me there isn't enough space — I have approximately 11 GB free, and it says I need 2 GB more.

Make sure you have the Current Version of iTunes Installed on your computer
iTunes free download from www.itunes.com/download
Then see Here  >  http://support.apple.com/kb/HT1808
You may need to try this More than Once...  Be sure to Follow ALL the Steps...
Take your time... Pay particular attention to Steps 3 and 4.
After you have Recovered your Device...
Re-Sync your Content or Restore from the most recent Backup...
Restore from Backup  >  http://support.apple.com/kb/ht1766
Note.  Once the Device is asking to be Restored with iTunes... it is too late to save anything... and you must continue with the Recovery...

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