1. **State the problem:** Visualize the feasible region defined by the constraint $-2x_1 + x_2 \leq 8$ and $x_1, x_2 \geq 0$. 2. **Rewrite the constraint:** $$x_2 \leq 8 + 2x_1$$ This is a line with slope 2 and y-intercept 8. 3. **Plot the boundary line:** - When $x_1 = 0$, $x_2 = 8$. - When $x_2 = 0$, solve $0 = 8 + 2x_1 \Rightarrow x_1 = -4$ (not in feasible region since $x_1 \geq 0$). 4. **Feasible region:** - Since $x_2 \leq 8 + 2x_1$ and $x_1, x_2 \geq 0$, the feasible region is the area below the line $x_2 = 8 + 2x_1$ in the first quadrant. - This region extends infinitely to the right and upwards. 5. **Objective function lines:** - Lines of constant $P = 2x_1 + 5x_2$ are straight lines with slope $-\frac{2}{5}$. - Increasing $P$ shifts these lines outward in the direction of vector $(2,5)$. 6. **Conclusion:** - The feasible region is unbounded. - The objective function increases without bound in the feasible region.