# Dual simplex method when initial reduced costs are negative

I have the following problem which I'm trying to solve by dual simplex method:

$$min -6x_1-14x_2-13x_3$$ s.t $$0.5x_1+2x_2+x_3 \le 24$$ $$x_1+2x_2+4x_3 \le 60$$ $$x_1+x_2 \ge 40$$ $$x_1, x_2, x_3 \ge 0$$

I change the constraints into equality as: $$0.5x_1+2x_2+x_3+s_1 = 24$$ $$x_1+2x_2+4x_3+s_2 = 60$$ $$-x_1-x_2+s_3=-40$$ The problem is that in the initial tableau the reduced costs are $$-6, -14, -13, 0, 0, 0$$ which violate the non-negativity clause for the reduced costs in dual simplex method.

Similarly there could be another type of problem with atleast 1 initial reduced cost as negative, such as:

$$min x_1-8x_2$$ s.t $$x_1+x_2 \ge 1$$ $$-x_1+6x_2 \le 3$$ $$x_1 \le 2$$ $$x_1, x_2, x_3 \ge 0$$

Again, I change the constraints into equality as: $$-x_1-x_2+s_1 = -1$$ $$-x_1+6x_2+s_2 = 3$$ $$x_1+s_3= 2$$ Here my initial reduced cost is $$(1, -8, 0, 0)$$. If I try solving the above two questions without taking the negativity of reduced costs into consideration, I stop at some intermediate solution when my $$B^{-1} b$$ becomes $$\ge 0$$ which is not optimal solution and my reduced vector is still not $$\ge 0$$. How should I then proceed with such problem if my reduced costs are negative?