First, let's simplify the expression step by step:

Simplify the equation: 20x + 0.4 * (-6 1/4) = 4 2/3 ÷ (-1/4)

Convert the mixed number -6 1/4 to an improper fraction: -6 1/4 = -25/4

Compute 0.4 * (-25/4) = -10

Rewrite the equation: 20x - 10 = 4 2/3 ÷ (-1/4)

Simplify the fraction 4 2/3 to an improper fraction: 4 2/3 = 14/3

Divide the fraction 14/3 by -1/4: (14/3) / (-1/4) = (14/3) * (-4/1) = -56/3

Rewrite the equation: 20x - 10 = -56/3

Add 10 to both sides of the equation: 20x - 10 + 10 = -56/3 + 10

Simplify the equation: 20x = -56/3 + 30/3 = -26/3

Divide both sides of the equation by 20: x = -26/3 ÷ 20 = -26/60 = -13/30

Therefore, the solution to the equation is x = -13/30 or -0.43333333333.