Originally Posted by iapetus
Yes, as choster said, all of the UA terminals are connected airside, so you won't have to clear security at LAX. If you have to go from terminal 8 to terminal 6, it might be something of a walk, which you'd have to do because there isn't a shuttle (as far as I know). But, you won't be arriving at terminal 8, since that's pretty much reserved for UX. So, a transfer between terminal 7 and terminal 6 is as far as you'd have to go.
Be careful when you go from 7 to 6 -- there is a lot of construction and the signs might not be clear. You will have to cross against a lot of people who are leaving terminal 6. They are going down the stairs to baggage claim - you don't want to do that!