In a small high school with 20 seniors, 8 of the seniors are in soccer, 9 of the seniors are in band, and 5 of the seniors are in both. How many seniors are in neither soccer nor band?

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The Venn diagram:

We start with 5 in the middle (both soccer and band). Since we know that 8 players are in soccer, we know that there must be 3 that play only soccer.

Since there are 9 total band members, there must be 4 that are only in band.

The number of students should add up to 20.

$$ \text{neither band or soccer} + 3 + 5 + 4 = 20 $$ $$ \text{neither band or soccer} + 12 = 20 $$ $$ \text{neither band or soccer} = \boxed{8} $$