Personal examples in TOEFL essays are often about friendships and relationships, so I often see sentences using “maintain relationships” and “keep relationships.”
How can students use these phrases properly?
This is the easiest one to use. It means to do what is necessary to continue in the relationship. Use it like this:
“It is important for us to maintain relationships with our old friends.”
“It can be challenging to maintain relationships with our friends when we go away to college.”
This one is a bit harder. It should include a pronoun and an adjective. Like this:
“It is important to keep our relationships strong.”
“Everyone should work hard to keep their relationships healthy.”
You can use “keep relationships” alone, but that has a meaning closer to “retain relationships” which sometimes sounds awkward, and isn’t usually the intended meaning.
Honestly, it is probably easiest to just use “Maintain” all the time. You can replace the above sentences with:
“It is important to maintain strong relationships.”
“Everyone should work hard to maintain healthy relationships.”
I think the above rules also apply to the word “friendships.”