How do you disagree without being disagreeable?
That is the problem with disagreeing with someone. However politely you try to express yourself, the other person may be determined to take offence.
Essentially, do not lose your temper, do not argue, use the Socratic question method to try and persuade the person to see the other side of the argument.
Not losing one's temper is truly essential. The "do not argue" thing is hard to do since you two have different opinions. Everyone has his own opinion. Do you have some range or something when you say "do not argue"? What is it's limits?
And could you please tell me how the Socratic questions method goes.
Rather than making statements, you ask the other person questions to lead them to the conclusion you want them to reach.
This site has a good explanation: Socratic questioning