Spacelines from the Far Out

First you need to get the health inspector to appear. The encounters appear to happen reliably when you have 1 or less star rating. If you need a quick way to tank your rating, just cut power to the gravity for a route and you will have no stars in no time. When selecting your next route, you should see an icon to indicate that there will be an inspector encounter.

Once you are on a route where a health inspector will appear, simply keep all the ships systems in working order, keep any messes cleaned up, and try to keep passengers happy so unexpected things do not happen. The inspector will scan any broken systems or messes and if the scan completes, you will fail the run. I think you can also dance to prevent them from scanning, but I never had to use this. The fewer passengers you have, the easier this is going to be, so I recommend trying to get these 5 inspections done as close to the start of a journey as possible if you are having any difficulty.