Do long-term capital gains get taxed directly at 15-20% and then the amount of income factor into what tax bracket I'm in?

Here's my issue. I'm making an investment that I will hold into 2019, thereby taking a long-term capital gain. I know I will pay a maximum tax of 20% on that income.

Where I'm hazy is what, if any, further impact does the capital gain have? Is it counted against my total income to push me into a higher bracket?

Example...

Say the long-term capital gain is $100,000.

Then, my wife and I earn $120,000 in salaries and for the sake of simplicity, say that our only deduction is the $24,000 standard deduction. That's a net of $96,000.

1.  So, we pay 15% for LTCG tax = $15,000

2.  Do we then pay 22% on the $96,000 net from our jobs, or is the capital gain somehow factored back in to drive us into the next bracket?

Please don't tell me to load everything into the software. This is speculative and I just need a direction.

Thanks.