A small business owner budgets $ 2,200 to purchase candles. The owner must purchase a minimum

The correct answer is $182$. Let $s$ represent the number of small candles the owner can purchase, and let $\mathcal{l}$ represent the number of large candles the owner can purchase. It’s given that the owner pays $\$4.90$ per candle to purchase small candles and $\$11.60$ per candle to purchase large candles. Therefore, the owner pays $4.90s$ dollars for $s$ small candles and $11.60\mathcal{l}$ dollars for $\mathcal{l}$ large candles, which means the owner pays a total of $4.90s+11.60\mathcal{l}$ dollars to purchase candles. It’s given that the owner budgets $\$\mathrm{2,200}$ to purchase candles. Therefore, $4.90s+11.60\mathcal{l}d\mathrm{2,200}$. It’s also given that the owner must purchase a minimum of $200$ candles. Therefore, $s+\mathcal{l}e200$. The inequalities $4.90s+11.60\mathcal{l}d\mathrm{2,200}$ and $s+\mathcal{l}e200$ can be combined into one compound inequality by rewriting the second inequality so that its left-hand side is equivalent to the left-hand side of the first inequality. Subtracting $\mathcal{l}$ from both sides of the inequality $s+\mathcal{l}e200$ yields $se200-\mathcal{l}$. Multiplying both sides of this inequality by $4.90$ yields $4.90se4.90\left(200-\mathcal{l}\right)$, or $4.90se980-4.90\mathcal{l}$. Adding $11.60\mathcal{l}$ to both sides of this inequality yields $4.90s+11.60\mathcal{l}e980-4.90\mathcal{l}+11.60\mathcal{l}$, or $4.90s+11.60\mathcal{l}e980+6.70\mathcal{l}$. This inequality can be combined with the inequality $4.90s+11.60\mathcal{l}d\mathrm{2,200}$, which yields the compound inequality $980+6.70\mathcal{l}d4.90s+11.60\mathcal{l}d\mathrm{2,200}$. It follows that $980+6.70\mathcal{l}d\mathrm{2,200}$. Subtracting $980$ from both sides of this inequality yields $6.70\mathcal{l}d\mathrm{2,200}$. Dividing both sides of this inequality by $6.70$ yields approximately $\mathcal{l}d182.09$. Since the number of large candles the owner purchases must be a whole number, the maximum number of large candles the owner can purchase is the largest whole number less than $182.09$, which is $182$.