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?

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.