photo library shows inconsistencies that need repair. How do I fix this without losing my photos?

First, make a backup copy of your photo library.

Are you using iPhoto or Photos?

In both applications, you can launch the Library First aid tools this way:

• Hold down the alt/options key and the command key at the same time. Double click the iPhoto library while holding down both keys firmly, until the first aid panel appears.
• In iPhoto select first to repair the permissions, then repeat with repair database. If you are using Photos, there will be only one repair option. Click "repair Library".