1. **Problem Statement:** (a) We want to graph the relationship between the number of hours Diego works out and the number of miles he walks per week. (b) We want to list three possible combinations of hours worked out and miles walked that meet Diego's goals. 8. **Problem Statement:** Emily wants to buy turquoise stones priced at 4 or 6 per stone, to give to at least 4 friends, spending no more than 30. (a) We want to graph the possible numbers of each price stone Emily can buy. (b) We want to list three possible solutions. --- 2. **Formulas and Constraints:** For Diego's workout and walking: - Let $x$ = hours worked out - Let $y$ = miles walked No explicit constraints given, so assume any non-negative values. For Emily's stones: - Let $x$ = number of 4-dollar stones - Let $y$ = number of 6-dollar stones - Emily wants to buy stones for at least 4 friends: $x + y \geq 4$ - Emily has no more than 30 to spend: $4x + 6y \leq 30$ - Also, $x \geq 0$, $y \geq 0$ --- 3. **Graphing Emily's stones:** Rewrite the inequality for spending: $$4x + 6y \leq 30$$ Solve for $y$: $$6y \leq 30 - 4x$$ $$y \leq \frac{30 - 4x}{6} = 5 - \frac{2x}{3}$$ Also, $x + y \geq 4$ means: $$y \geq 4 - x$$ So the feasible region is bounded by: $$y \leq 5 - \frac{2x}{3}$$ $$y \geq 4 - x$$ $$x \geq 0, y \geq 0$$ --- 4. **Three possible solutions for Emily:** Try integer values satisfying constraints: - $x=3$, $y=1$: $4(3) + 6(1) = 12 + 6 = 18 \leq 30$, and $3 + 1 = 4 \geq 4$ - $x=1$, $y=4$: $4(1) + 6(4) = 4 + 24 = 28 \leq 30$, and $1 + 4 = 5 \geq 4$ - $x=5$, $y=0$: $4(5) + 6(0) = 20 \leq 30$, and $5 + 0 = 5 \geq 4$ --- 5. **Diego's workout and walking graph:** Since no explicit constraints or function given, the graph can be any points $(x,y)$ where $x,y \geq 0$. Three possible combinations (hours, miles): - (2, 3) - (4, 1) - (1, 5) --- **Final answers:** - Diego's combinations: (2,3), (4,1), (1,5) - Emily's combinations: (3,1), (1,4), (5,0)