To solve this equation, we need to set each factor equal to zero and solve for x.

First, setting |x| - 3/7 equal to zero:
|x| - 3/7 = 0
|x| = 3/7

This gives us two equations to consider:
1) x = 3/7
2) x = -3/7

Next, setting 4/7x + 8 equal to zero:
4/7x + 8 = 0
4/7x = -8
x = -8 * 7/4
x = -14

Therefore, the solutions to the equation are x = 3/7, x = -3/7, and x = -14.