A saline solution is 20% salt. How many gallons of water must be added to dilute the mixture to 8gal of a.....?

0% .... 15% ... 20%
  ... 3 .... .... 1 ..... (distances in units of 5% of the ingredients from the desired concentration)

Reverse the distances to get the proportions:
3 + 1 = 4,
so 3/4 will be the 20% solution, and 1/4 will be the water.

3/4 * 8 gallons = 6 gallons of 20% saline
1/4 * 8 gallons= 2 gallons of water

Result is:
6 gallons of 20% solution = 1.2 gallons of pure salt
1.2 / 8 = 15%, as desired.

Answer:
6 gallons of 20% solution + 2 gallons of 0% solution (i.e. plain water) is what you want.

That is my preferred method of doing these mixture problems,
but you can also work backwards:

15% of 8 gallons = 1.2 gallons of pure salt
1.2 / 0.20 = 6 gallons of 20% saline solution
8 - 6 = 2 gallons of water.