Which one is correct? (Present Perfect Continuous vs Present Perfect)

1. My sister and her boyfriend have not been going out together for a long time.

Or

2. My sister and her boyfriend have not gone out together for a long time.

Answer

Both are correct. They mean different things.

1. My sister and her boyfriend have not been going out together for a long time.

They stopped dating each other a long time ago (to be going out = to be dating each other).

2. My sister and her boyfriend have not gone out together for a long time.

They are, quite possibly, still 'dating' but they have not actually been out together (on a date) for a long time. (to go out = (er...) to go out!)