1. **State the problem:** We want to solve the linear programming problem using the simplex method: Maximize $$Z = 3x_1 + 2x_2$$ subject to constraints: $$2x_1 + x_2 \leq 40$$ $$x_1 + x_2 \leq 24$$ $$2x_1 + 3x_2 \leq 60$$ with $$x_1, x_2 \geq 0$$. 2. **Convert inequalities to equalities by adding slack variables:** $$2x_1 + x_2 + s_1 = 40$$ $$x_1 + x_2 + s_2 = 24$$ $$2x_1 + 3x_2 + s_3 = 60$$ where $$s_1, s_2, s_3 \geq 0$$ are slack variables. 3. **Set up the initial simplex tableau:** \begin{array}{c|ccccc|c} & x_1 & x_2 & s_1 & s_2 & s_3 & \text{RHS} \\ \hline s_1 & 2 & 1 & 1 & 0 & 0 & 40 \\ s_2 & 1 & 1 & 0 & 1 & 0 & 24 \\ s_3 & 2 & 3 & 0 & 0 & 1 & 60 \\ \hline -Z & -3 & -2 & 0 & 0 & 0 & 0 \end{array} 4. **Initial basic feasible solution:** Slack variables $$s_1, s_2, s_3$$ are basic with values 40, 24, and 60 respectively. 5. **Dual Simplex method setup:** The dual simplex method is used when the initial solution is infeasible but optimality conditions hold. Here, the initial solution is feasible, so normally we use the primal simplex. However, if requested, the dual simplex tableau is similar but we look for negative RHS values to pivot. Since all RHS values are positive, the initial tableau is feasible for primal simplex. 6. **Summary:** The initial tableau for the simplex method is: $$\begin{array}{c|ccccc|c} & x_1 & x_2 & s_1 & s_2 & s_3 & \text{RHS} \\ \hline s_1 & 2 & 1 & 1 & 0 & 0 & 40 \\ s_2 & 1 & 1 & 0 & 1 & 0 & 24 \\ s_3 & 2 & 3 & 0 & 0 & 1 & 60 \\ \hline -Z & -3 & -2 & 0 & 0 & 0 & 0 \end{array}$$ This tableau can be used to proceed with the primal simplex method to find the optimal solution. **Final answer:** The initial simplex tableau is set up as above for the given problem.