Well, it could be possible to run all of it, but funnily enough, run/walk may get you a faster time.
So it rather depends on whether you're interested in running the whole thing, or chalking up a personal best
Anyway, if you have 3/4 outings per week, I think you can probably run it and keep run/walk as a contingency, if you want.
The good news is that you only need to increase distance on one run per week. If you think about how you are increasing your workload, what you want to do is accustom your body to time on your feet and time spent operating with a raised heart rate. So I think I would suggest establishing one day a week where you can set aside up to 2 hours for a long slow run. To try and build up the time on your feet you might want to break the run up with a walk period.
So for the next 5 weeks your long slow runs might look like this (assuming they're on a sunday)
Sun 14 run 3 miles, walk for 10 minutes, run 2 miles
Sun 21 run 5 miles, walk for 10 minutes, run 2 miles
Sun 28 run 6 miles, walk for 5 minutes, run 2 miles
Sun 4 run 8 miles
Sun 11 run 8 miles and either walk for 10 minutes and try another 2 or just try for one more mile.
Remember: you must run these runs much slower than your 3 miler. I don't care if it feels like you're jogging, it's the only way you're going to do it safely . If you can't sing a (rather panty) version of 'The Grand Old Duke of York' you're going too fast. Sounds tedious I know but we're increasing the distance faster than is recommended here and if we're not careful it could damage you.
On the same note, for the next couple of weeks you might want to stick at three outings a week. It always helps to be able to go a bit faster, so you can try using your three mile run as the basis for some speed games (jog first mile, run 3 x 1 minute a bit faster in the second mile, jog home again, sort of thing). For the third session you could try to run the same distance as the first block of your long slow run perhaps.
You'll need to be sure to leave at least one rest day between each run and I would leave two before your long slow run.
If you're still feeling strong after two weeks of three outings a week, then we can try to add one more small session in, which will be at the same pace as the long slow run, but shorter (probably another three miler).
So, to clarify, a typical three outing per week schedule would look like this:
Sun - long slow run
Mon - rest and eat biscuits
Tues - 3 miles with some faster bits and proper warm up/warm down
Wed - rest
Thurs - first block of your long slow run, but a tiny bit faster.
Fri - rest
Sat - rest
And listen to your body - if you have mild pain that lasts more than 5 mins when exercising or a severe pain, stop and walk. If you are sitting or standing still and there is a perceptible pain, then something isn't right and you should think about skipping or deferring any outings.
Sorry, that's a bit of an essay, but hopefully there will some useful bits in it!