> So if you have a widget and it matches claim 1-8 but does not match claim 9, it does not match.

This is not correct. If you infringe claim 1 then you infringe claim 1. Whether or not you also infringe claim 9 doesn't matter.

Assuming Claim 9 is dependent on claim 1, claim 9 includes all the limitations of claim 1. So claim 1 is A+B+C.... Claim 9 is 1 + D = A+B+C+D. So if you infringe 9 because you have A+B+C+D then you also, by definition, infringe claim 1 because you have A+B+C.

Also, you can write a claim that has ORs applied. But the drafter has to be really careful with how it is done and it is not the norm.