UX to UA is all in the same terminal complex at LAX (Terminals 6-8) and transfers are behind security if you keep your eyes open and follow the signs.

I would never chance a connection to an international departure for under an hour. 90-120 minutes is my comfort level. I'm going to assume you'll have checked luggage going to SYD; I would have doubts that UA could get the luggage off the SLC flight and over to the SYD flight. You'd need everything to work right and have no margin for error.

I'd look for some better flights, maybe via SFO and/or through DEN to make a longer connection and then ask UA to make that change.

If not, I would make some contingency plans. I would sit as far forward on the SLC flight as I can so I can get off the plane in quick order, I would ask the F/A what the arrival gate will be at LAX, I would look at a map of LAX and figure out how to get from my arrival gate to the end of Terminal 7 (where the SYD flight departs), and I would pack some clothes in a small duffle that I could carry on the SJC flight without having to gate check it. You don't want to gate check anything in SLC and have to wait for it in LAX.