1. **State the problem:** We want to minimize the objective function $$Z = 3x_1 + 4x_2$$ subject to the constraints: $$4x_1 + x_2 \geq 30$$ $$-x_1 - x_2 \leq -18$$ $$x_1 + 3x_2 \geq 28$$ and $$x_1, x_2 \geq 0$$. 2. **Form the dual problem:** The primal is a minimization with mixed inequality constraints. We convert it to the dual maximization problem. - Let dual variables be $$y_1, y_2, y_3$$ corresponding to each constraint. - For $$\geq$$ constraints, dual variables $$y_1, y_3 \geq 0$$. - For $$\leq$$ constraint, dual variable $$y_2 \leq 0$$. The dual objective is: $$\max W = 30y_1 + 18y_2 + 28y_3$$ Dual constraints come from coefficients of $$x_1$$ and $$x_2$$: $$4y_1 - y_2 + y_3 \leq 3$$ $$y_1 - y_2 + 3y_3 \leq 4$$ 3. **Summarize dual problem:** $$\max W = 30y_1 + 18y_2 + 28y_3$$ subject to $$4y_1 - y_2 + y_3 \leq 3$$ $$y_1 - y_2 + 3y_3 \leq 4$$ with $$y_1, y_3 \geq 0, y_2 \leq 0$$. 4. **Solve the dual problem:** We try to find values of $$y_1, y_2, y_3$$ satisfying constraints and maximizing $$W$$. From constraints: - $$4y_1 - y_2 + y_3 \leq 3$$ - $$y_1 - y_2 + 3y_3 \leq 4$$ Since $$y_2 \leq 0$$, let $$y_2 = -t$$ with $$t \geq 0$$. Rewrite constraints: - $$4y_1 + t + y_3 \leq 3$$ - $$y_1 + t + 3y_3 \leq 4$$ Objective: $$W = 30y_1 - 18t + 28y_3$$ Try to maximize $$W$$ under these constraints and non-negativity of $$y_1, y_3, t$$. 5. **Check corner points:** Try $$t=0$$ for simplicity: - $$4y_1 + y_3 \leq 3$$ - $$y_1 + 3y_3 \leq 4$$ Maximize $$W = 30y_1 + 28y_3$$. Check intersection: From first: $$y_3 = 3 - 4y_1$$ Substitute into second: $$y_1 + 3(3 - 4y_1) \leq 4 \Rightarrow y_1 + 9 - 12y_1 \leq 4 \Rightarrow -11y_1 \leq -5 \Rightarrow y_1 \geq \frac{5}{11}$$ At $$y_1 = \frac{5}{11}$$, $$y_3 = 3 - 4 \times \frac{5}{11} = 3 - \frac{20}{11} = \frac{13}{11}$$. Calculate $$W$$: $$W = 30 \times \frac{5}{11} + 28 \times \frac{13}{11} = \frac{150}{11} + \frac{364}{11} = \frac{514}{11} \approx 46.73$$. 6. **Check feasibility and non-negativity:** $$y_1 = \frac{5}{11} > 0, y_3 = \frac{13}{11} > 0, t=0 \Rightarrow y_2 = 0$$. All conditions satisfied. 7. **Dual optimal value:** $$W^* = 46.73$$. 8. **By strong duality, primal minimum:** $$Z^* = W^* = 46.73$$. **Final answer:** The minimum value of $$Z$$ is approximately $$46.73$$.