Work out puts more emphasis on people's problem-solving behavior, and solve problem puts more emphasis on results, that is, the problem has been solved.

This will not solve the problem, but it is a step in the right direction.

Although this will not solve the problem, it is a step in the right direction.

It is doubtful whether this is a good way to solve the problem.

This is not necessarily a good way to solve the problem.

Go to the gym to exercise or swim twice a week.

Go to the gym twice a week, or swim twice.

While the boys are playing golf, I exercise in the gym.

While the boys are playing golf, I exercise in the gym.