https://file.aiquickdraw.com/mathbot/file/1730407490476igq3a48z.jpg

This problem involves linear programming, where you need to graph the feasible region and the objective function based on the provided constraints.

### Problem Analysis

The objective function is:
\[
\text{Maximize } z = x_1
\]

Subject to these constraints:
1. \(-x_1 + x_2 \leq 2\)
2. \(x_1 + x_2 \leq 8\)
3. \(-x_1 + x_2 \geq -4\)
4. \(x_1 \geq 0\)
5. \(x_2 \geq 0\)

### Steps to Graph

1. **Rewrite each inequality constraint as an equation to find boundary lines.**
   - For \(-x_1 + x_2 = 2\): this line will pass through points like \((0, 2)\) and \((2, 4)\).
   - For \(x_1 + x_2 = 8\): this line passes through points like \((0, 8)\) and \((8, 0)\).
   - For \(-x_1 + x_2 = -4\): this line passes through points like \((0, -4)\) and \((4, 0)\).
   - The conditions \(x_1 \geq 0\) and \(x_2 \geq 0\) ensure the feasible region is in the first quadrant.

2. **Shade the feasible region**:
   - For each inequality, shade the area that satisfies the condition, typically by testing points.
   - The feasible region will be the intersection of all shaded areas.

3. **Graph the objective function**:
   - To maximize \(z = x_1\), select different values of \(z\) (e.g., \(z = 1\), \(z = 2\)) to plot lines \(x_1 = 1\), \(x_1 = 2\), etc.
   - The goal is to identify the boundary of the feasible region where \(x_1\) is maximized.

Would you like a detailed graph of the feasible region and objective function steps?

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### Further Questions for Practice

1. How do you interpret the feasible region in a linear programming problem?
2. What happens to the objective function line as you increase \(z\)?
3. How can you verify that a solution is optimal?
4. What role do the constraints \(x_1 \geq 0\) and \(x_2 \geq 0\) play in defining the feasible region?
5. Why does maximizing \(z = x_1\) involve finding the largest \(x_1\) within the feasible region?

#### Tip
In linear programming, always check for intersection points of boundary lines, as these often provide optimal solutions.

Consider the linear program: Maximize z = x1, subject to: -x1 + x2 ≤ 2, x1 + x2 ≤ 8, -x1 + x2 ≥ -4, x1 ≥ 0, x2 ≥ 0. a) Graph the feasible region. b) Graph the objective function.

Learn how to solve a linear programming problem by graphing the feasible region and objective function. This guide covers the steps for maximizing z = x1 subject to constraints, finding boundary lines, and determining the optimal solution.

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