This is a linear programming problem. The goal is to find the minimum value of the objective function $Z = 4x_1 + x_2$, subject to several constraints:

Problem Summary:

Objective Function: $\text{Minimize } Z = 4x_1 + x_2$

Constraints:

1. $3x_1 + x_2 = 3$
2. $4x_1 + 3x_2 \geq 6$
3. $x_1 + 2x_2 \leq 4$
4. $x_1, x_2 \geq 0$

Approach:

1. Identify Feasible Region: The feasible region is determined by the constraints. We need to find the intersection of all constraints.
2. Determine Corner Points: In linear programming, the optimal solution (minimum or maximum) generally occurs at the vertices (corner points) of the feasible region.
3. Evaluate Objective Function: Once the corner points are found, we substitute them into the objective function to find the minimum value.

Step-by-Step Solution:

1. Express constraints as equations:

   • $3x_1 + x_2 = 3$
   • $4x_1 + 3x_2 \geq 6$
   • $x_1 + 2x_2 \leq 4$
   • $x_1 \geq 0$ and $x_2 \geq 0$

2. Identify intersections: We find intersections between these lines, which will serve as corner points for the feasible region.

3. Calculate the intersection points by solving these equations, and check which ones satisfy all constraints. For example:

   • Solve $3x_1 + x_2 = 3$ and $4x_1 + 3x_2 = 6$.
   • Solve $3x_1 + x_2 = 3$ and $x_1 + 2x_2 = 4$.
   • Solve $x_1 + 2x_2 = 4$ and $x_1 = 0$, etc.

4. Evaluate $Z$ at these intersection points to find the minimum value.

Would you like me to continue with the specific calculations for each step, including finding intersections and evaluating the objective function?

Here are five relative questions to explore further:

1. What is the role of the objective function in linear programming?
2. How are corner points relevant to solving linear programming problems?
3. What happens if the feasible region is unbounded?
4. How do we verify if a point satisfies all constraints in a linear programming problem?
5. What is the difference between a minimization and maximization problem in linear programming?

Tip: For solving linear programming problems graphically, always focus on plotting constraints accurately to identify the feasible region.