I think except for
Bunuel all others are wrong, they simply take the 3z to the other side of the inequality without knowing whether z is positive or negative. Sorry If I am rude, but taking a variable to the other side of the inequality without knowing its sign is just plain wrong. If z were positive there is no change in the inequality sign, but if z were negative we must change the direction of the inequality. Thus the approach of adding two equations seem to be flawless.
So please don't take the z to the other side, or in general do not take any variable to the other side of the inequalilty without knowing it's sign, this affects the equation as a whole. Try and make a rational decision.
You should carefully note that bunuel took the z to the other side, only after "he proved z to be greater than 0". By adding two equations which were already greater than zero individually.