Tell me please if I need to use check or check on in the following context.
I am not sure if the facts he cited are true, so I had better check/check on them.
If both are correct there, then what is the difference?
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Check on X means make sure X is OK.
Check X means to get the current state of X, but you typically wouldn't do something unless X is doing something wrong. Because of that implication it's often interchangeable with check on.
Check on X is probably a bit more typically used if X is some distance from you or some time will pass before you actually check.
Check on the food in an hour.
Check the food now.
but switching these around isn't incorrect and doesn't sound strange.
In the sense of going to someone's house and making sure someone is OK, you'd almost always use check on and not check. Not using on in the following sentences sounds a bit like you mean "check" in the sense of "medically examine":
Check on Margaret when you get into town.
Check on my cat when you get home.
Facts are not people, so you'd say check and not check on.