I'm a beginner and I'm working on a bike. I'm trying to fix the shifting gears. I'm holding the shifter cable where it comes out of the housing by the front derailleur and I feel no change in tension from gears 3-5. At 1 and 2 I do feel a change and it is difficult to shift to those positions. It's a shimano grip shifter. I'm following Park Tools "How to adjust a derailleur" but am hitting a problem shifting from 5 to 4. It doesn't shift and no amount of rotations of the barrel adjuster seems to resolve. I just noticed that if I hold the brake cable where it comes out of the shift I do feel the tension change.
Is this a problem in the grip shifter? It seems backwards, shouldn't 4-5 be difficult to shift, as that should tighten the shift cable? I did take the shifter off to inspect so maybe I didn't put it back on right. It seems kind of dummy proof as I only saw one way to put it back together.
I've added a pic of the inside of the grip shifter. Here is another noob question. There are only 3 teeth, how do three teeth make for five gears?
Thanks for any help!