1. **Problem Statement:** Mary makes $n$ bracelets each week and sells them for $19 - n$ dollars each. Her costs include a fixed cost of 25 plus 2.50 per bracelet. We need to write equations for total weekly cost and revenue, then find how many bracelets she should make to make a profit. 2. **Write the cost equation:** Total cost $C$ is fixed cost plus variable cost: $$C = 25 + 2.50n$$ 3. **Write the revenue equation:** Revenue $R$ is number of bracelets times price per bracelet: $$R = n(19 - n) = 19n - n^2$$ 4. **Find profit condition:** Profit occurs when revenue exceeds cost: $$R > C$$ Substitute equations: $$19n - n^2 > 25 + 2.50n$$ 5. **Simplify inequality:** $$19n - n^2 - 2.50n > 25$$ $$16.5n - n^2 > 25$$ Rewrite: $$-n^2 + 16.5n - 25 > 0$$ Multiply both sides by $-1$ (reverse inequality): $$\cancel{-}n^2 + \cancel{16.5}n - \cancel{25} < 0$$ $$n^2 - 16.5n + 25 < 0$$ 6. **Solve quadratic inequality:** Find roots of $n^2 - 16.5n + 25 = 0$ using quadratic formula: $$n = \frac{16.5 \pm \sqrt{16.5^2 - 4 \times 1 \times 25}}{2}$$ Calculate discriminant: $$16.5^2 = 272.25$$ $$\sqrt{272.25 - 100} = \sqrt{172.25} = 13.125$$ Roots: $$n = \frac{16.5 \pm 13.125}{2}$$ So, $$n_1 = \frac{16.5 - 13.125}{2} = \frac{3.375}{2} = 1.6875$$ $$n_2 = \frac{16.5 + 13.125}{2} = \frac{29.625}{2} = 14.8125$$ 7. **Interpret inequality:** Since parabola opens upward, $n^2 - 16.5n + 25 < 0$ between roots: $$1.6875 < n < 14.8125$$ 8. **Conclusion:** Mary should make between 2 and 14 bracelets (whole numbers) to make a profit. **Final answers:** - Cost equation: $$C = 25 + 2.50n$$ - Revenue equation: $$R = n(19 - n)$$ - Profit when $$1.6875 < n < 14.8125$$, so $n$ is 2 to 14 bracelets.