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LA is a nightmare without a car. If you work near a rail stop and want to live near another one, you better be prepared to pay out the ass for it unless it's in the ghetto. If you're willing to drive/bus to a train station and then ride downtown or wherever, your best bet is probably somewhere off the Red Line in the SFV or somewhere the north side of the Gold Line.
That said pretty much entirely avoid anything on the blue line between the Long Beach loop and Grand, and just stay away from the whole green line altogether unless you're going to LAX. Parts of the gold line are kind of sketchy too. (Most of the stations near downtown.)