1. **Stating the problem:** Melanie wants to model constraints for cycling and aerobic stepping using linear programming. 2. **Define variables:** Let $x$ = hours spent cycling. Let $y$ = hours spent aerobic stepping. 3. **Constraints given:** - At least 9 hours total: $$x + y \geq 9$$ - At least 2 hours cycling: $$x \geq 2$$ - Total hours cannot exceed 24: $$x + y \leq 24$$ - At least 3 times aerobic stepping compared to cycling: $$y \geq 3x$$ 4. **Graphing the constraints:** - The line $x + y = 9$ divides the plane; feasible region is above or on this line. - The line $x = 2$ is vertical; feasible region is to the right or on this line. - The line $x + y = 24$ divides the plane; feasible region is below or on this line. - The line $y = 3x$ divides the plane; feasible region is above or on this line. 5. **Summary of feasible region:** The feasible region is the intersection of all these inequalities. 6. **Example of checking a point:** Check if $(x,y) = (3,9)$ satisfies all: - $3 + 9 = 12 \geq 9$ ✓ - $3 \geq 2$ ✓ - $3 + 9 = 12 \leq 24$ ✓ - $9 \geq 3 \times 3 = 9$ ✓ 7. **Final model:** $$\begin{cases} x + y \geq 9 \\ x \geq 2 \\ x + y \leq 24 \\ y \geq 3x \end{cases}$$ This system can be graphed to find the feasible region representing all possible combinations of cycling and aerobic stepping hours Melanie can do under the constraints.