1. **State the problem:** Lucy exercises weekly with two types of activities: cardiovascular work ($x$ hours) and weight training ($y$ hours). She has these constraints: - Total exercise time is at least 9 hours: $$x + y \geq 9$$ - Cardiovascular work is at most 11 hours: $$x \leq 11$$ - Weight training is at most 5 hours: $$y \leq 5$$ - Both $x$ and $y$ must be non-negative since time cannot be negative: $$x \geq 0, y \geq 0$$ 2. **Explain the inequalities:** - The inequality $$x + y \geq 9$$ means the combined time of both exercises is at least 9 hours. - The inequalities $$x \leq 11$$ and $$y \leq 5$$ limit the maximum hours spent on each activity. - The non-negativity constraints ensure the time values are realistic. 3. **Graphing the region:** - The feasible region is the set of all points $(x,y)$ that satisfy all inequalities simultaneously. - This region lies above or on the line $$x + y = 9$$. - It is bounded on the right by the vertical line $$x = 11$$. - It is bounded below by the horizontal axis $$y = 0$$ and on top by $$y = 5$$. - It is bounded on the left by $$x = 0$$. 4. **Summary:** The shaded region includes all points $(x,y)$ where: $$0 \leq x \leq 11$$ $$0 \leq y \leq 5$$ $$x + y \geq 9$$ This region represents all possible combinations of cardiovascular and weight training hours Lucy can do to meet her weekly exercise goals.