1. **State the problem:** We are given a system of inequalities: - $x \geq 3$ - $y \leq 9$ - $x + y \leq 14$ - $8x + 15y \geq 150$ We want to understand the feasible region defined by these inequalities and verify that the point $(5,8)$ is a solution. 2. **Explain the inequalities:** - $x \geq 3$ means all points to the right of or on the vertical line $x=3$. - $y \leq 9$ means all points below or on the horizontal line $y=9$. - $x + y \leq 14$ means all points on or below the line $y = 14 - x$. - $8x + 15y \geq 150$ means all points on or above the line $y = \frac{150 - 8x}{15}$. 3. **Check the point $(5,8)$:** - Check $x \geq 3$: $5 \geq 3$ is true. - Check $y \leq 9$: $8 \leq 9$ is true. - Check $x + y \leq 14$: $5 + 8 = 13 \leq 14$ is true. - Check $8x + 15y \geq 150$: $8(5) + 15(8) = 40 + 120 = 160 \geq 150$ is true. All inequalities are satisfied by $(5,8)$. 4. **Summary:** The feasible region is the intersection of all these inequalities, which forms a polygon bounded by the lines $x=3$, $y=9$, $x+y=14$, and $8x+15y=150$. The point $(5,8)$ lies inside this region.