1. **Problem Statement:** We want to find how many t-shirts and tickets must be sold to raise at least 5000 for charity. 2. **Identify variables:** Let $x$ = number of t-shirts sold Let $y$ = number of tickets sold 3. **Write the linear inequality:** Each t-shirt contributes 10 to charity, each ticket contributes 32. The total amount raised must be at least 5000. So, the inequality is: $$10x + 32y \geq 5000$$ 4. **Restrictions on variables:** Since $x$ and $y$ represent quantities sold, they must be non-negative integers: $$x \geq 0, \quad y \geq 0$$ 5. **Graphing the inequality:** The boundary line is: $$10x + 32y = 5000$$ To graph: - When $x=0$, solve for $y$: $$32y = 5000 \Rightarrow y = \frac{5000}{32} = 156.25$$ - When $y=0$, solve for $x$: $$10x = 5000 \Rightarrow x = 500$$ The solution region is the area above or on this line (since $\geq$). 6. **Check points:** (i) $(400, 20)$: $$10(400) + 32(20) = 4000 + 640 = 4640 < 5000$$ Not in the solution set. (ii) $(205, 98)$: $$10(205) + 32(98) = 2050 + 3136 = 5186 \geq 5000$$ In the solution set. (iii) $(156, 105)$: $$10(156) + 32(105) = 1560 + 3360 = 4920 < 5000$$ Not in the solution set. **Final answers:** - Inequality: $$10x + 32y \geq 5000$$ - Restrictions: $$x \geq 0, y \geq 0$$ - Points in solution set: only $(205, 98)$