Sending by distant is when I put my hands close together in almost a prayer-like postion (but not touching) and I visualize the people or home in between my hands. When "I ask that they be able to work out their problems so that we all can live in harmony." is stating my intent or purpose rather. The way I was taught is that reiki is best when you make your intent and purpose clear. So by me asking that they be able to work out their problems, they may have had a bad day at work, worrying about money or whatever the case may be which makes them unhappy and negative. Well when people are unhappy and negative, other people feel that because they are often rude to others and treat them badly which makes it difficult to live around.
So by being compassionate and sending them reiki energy to help them work out their problems, when they do work them out, they are more pleasant to others and easier to live around.
I hope that helps answer your question. :)