To 'tease out something' is a phrasal verb meaning to try to obtain information or to understand something which is not clear. When you 'tease out a solution' or 'tease out a problem' it means you find a solution even though it is difficult. To tease out something can also mean to separate something gradually, such as a knot in your hair, tangles or something which is stuck together.

1. Tom didn't want anyone to know the date of his wedding just yet, but through clever questioning, I managed to tease it out of him.
2. Janet kept insisting that she hadn't done anything wrong, but Gary wouldn't let it go, and teased the truth out of her in the end.

1. The report I was handed was so long and poorly written that it took me a long time to tease out the relevant details.
2. In his role as mediator, Luke was tasked with teasing out a solution to the ongoing industrial dispute. He warned that it could take quite some time.