Sure. My need to explain it shows you have no ATC experience and hence have been telling porkies.
The a/c in that scenario are about 3 nm apart, and are at FL410. At this point, you are pretty much screwed. The only possible thing that you might be able to get out, in the time available, is to turn the 'northbound' to the right to pass behind the other (by a very small margin, and only if the pilot were to act immediately. If this were a real scenario, the pilots would already be getting RAs). Airline a/c maneuver very slowly at high altitude, so the geometry of this scenario dictates that turning the eastbound right, and the northbound left (which is your question, correct?) would only ensure the conflict worsens. For you solution to be a valid way out, you need to have enough room (time) available so that the a/c can turn away from each other BEFORE the northbound crosses the westbound's projected path, and vice versa. Does that make sense?
If you don't have that time, then you help by trying to stretch the time available before one crosses the others path. This is achieved by turning them BOTH the SAME direction (as described in the advice).