How/why is switch made from equation to inequality for certificate of optimality for dual problem in linear programming?
If the required conditions are met then you have
$$ x_1 + 6x_2 \leq (y_1+y_3)x_1 + (y_2+y_3)x_2 = y_1 x_1 + y_2 x_2 + y_3(x_1 + x_2) \leq 200y_1 + 300y_2 + 400y_3$$
By looking at the leftmost and rightmost expressions you get the inequality you want.