You and a friend are discussing your New Year's resolutions with each other, a few days after New Year's Day. You always work late hours, and you don't get to see your family very much. This year, you'd like to see them more often. You tell your friend:
I'm going to make an effort to spend more quality time with my family.
This means to "try" to do something:
I've made an effort to watch what I eat a little more carefully in the past few weeks.
You usually use it when you haven't been successful doing something, or you don't think you're going to be successful. In the example above, the speaker say that he's going to "make an effort" to spend more time with his family. That might mean that he's not really sure whether he'll be able to do that.
"Quality time" is time that you spend with your family members, someone you're in a relationship with, or really close friends. It's time that you spend doing things together, like talking, playing sports together, or going out somewhere together. If you're talking to a family member and watching TV or doing work at the same time, that's not "quality time". You have to be able to concentrate on the other person.
In American culture, family members are expected to "spend quality time" with each other. If parents don't spend quality time with their children, the children may feel like the parents don't love them. Husbands and wives, as well as serious boyfriends and girlfriends, are also supposed to spend quality time with each other.
It's a common New Year's resolution to say that you want to spend more time with your friends and family.
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