Erase and Sync??

So I am trying to add music from a CD onto my mom's iphone and it is telling me that it is synced to another library... Problem is that it has never been plugged into any computer other than mine.

I have old backups of the phone but I do not want to "erase and sync" because I am worried that something might get lost.

What can I do to fix this? I do not want to use a third party software.

Thanks for the help!