It means to do something for someone because you care about them. For example, "Max pushed Sam out of the way of a falling piano, and Sam landed hard on the ground, scraping her knee. Max didn't mean to hurt Sam. He was acting out of concern for Sam's well-being.
act out of concern for somebody 
For example, "Max pushed Sam out of the way of a falling piano, and Sam landed hard on the ground, scraping her knee. Max didn't mean to hurt Sam. He was acting out of concern for Sam's well-being. 
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