https://file.aiquickdraw.com/mathbot/file/1730270097771xphi17j6.jpg

This is a linear programming problem aiming to minimize the objective function \( Z = 3x_1 + 4x_2 \), subject to a set of constraints.

### Problem Details:

1. **Objective Function:**
   \[
   \text{Minimize } Z = 3x_1 + 4x_2
   \]

2. **Subject to Constraints:**
   \[
   x_1 + x_2 = 10
   \]
   \[
   x_1 \geq 5
   \]
   \[
   x_2 \geq 4
   \]
   \[
   x_1, x_2 \geq 0
   \]

### Solution Approach:
To solve this, we can use the graphical method or simplex method, as it's a linear program with two variables. Here's the step-by-step approach for the graphical method:

1. **Identify Feasible Region:**
   - Plot the constraints on a coordinate plane.
   - Determine the feasible region that satisfies all inequalities.

2. **Find Intersection Points:**
   - Solve the equations to find where the constraint lines intersect.
   - Only consider points in the feasible region.

3. **Evaluate Objective Function:**
   - Substitute feasible points (vertices of the feasible region) into the objective function to find the minimum value of \( Z \).

Would you like a detailed solution using these steps? 

### Further Questions
1. How do you plot the feasible region on a graph for linear programming problems?
2. What are the typical methods for solving linear programming problems, and when are they used?
3. What role do boundary points play in finding the optimal solution in linear programming?
4. Can the simplex method be applied directly to this problem, or is it better suited for graphical analysis?
5. How do we handle constraints with equality or inequality signs in linear programming?

### Tip:
In linear programming problems, always identify the feasible region first, as the optimal solution lies at one of the vertices of this region.

Minimize Z = 3x1 + 4x2 subject to constraints: x1 + x2 = 10, x1 ≥ 5, x2 ≥ 4, x1, x2 ≥ 0.

Explore the solution to a linear programming problem aiming to minimize the function Z = 3x1 + 4x2 under specific constraints. This exercise involves optimization, graphical methods, and identifying feasible regions.

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