We were asked to solve the problem 1/x < 4.

I solved it like so:
1<4x
1/4<x
and wrote my answer in interval notation as (1/4, infinity). However, the answer in the back of my book said the answer was the union between (negative infinity, 0) and (1/4, infinity). How do you find the other solution set?

Your first step was incorrect. You can't simply multiply both sides by x and keep the < sign in the same direction, because its direction depends upon the sign of x.

If x>0, then 1 < 4x amd x > 1/4
if x < 0, then 1 > 4x and x < 1/4, which is guaranteed since we assumed x <0. Therefore x is either any number from -infinity to 0, or from 1/4 to +infinity.