1. **State the problem:** Kofi's bank balance in September varies according to an absolute value function. He started with 1600, had 100 on the 9th, and the lowest balance was -900. We want to find for how many days in September his balance was at least 600. 2. **Define the function:** Let $B(t)$ be the balance on day $t$ of September, where $t=1$ is September 1. Since the balance varies as an absolute value function, we model it as: $$B(t) = a|t - h| + k$$ where $a$, $h$, and $k$ are constants. 3. **Use given points:** - On day 1, $B(1) = 1600$ - On day 9, $B(9) = 100$ - The minimum balance is $-900$, which occurs at $t = h$ (the vertex of the absolute value function), so $B(h) = -900$ 4. **Find $a$, $h$, and $k$:** Since $B(h) = k = -900$, the vertex is at $(h, -900)$. Using $B(1) = 1600$: $$1600 = a|1 - h| - 900$$ $$a|1 - h| = 2500$$ Using $B(9) = 100$: $$100 = a|9 - h| - 900$$ $$a|9 - h| = 1000$$ 5. **Set up ratio to find $h$:** $$\frac{|1 - h|}{|9 - h|} = \frac{2500/a}{1000/a} = \frac{2500}{1000} = 2.5$$ 6. **Solve for $h$:** Consider two cases for $h$ between 1 and 9: - Case 1: $h < 1$ or $h > 9$ (unlikely since minimum should be between given points) - Case 2: $1 < h < 9$ Assuming $1 < h < 9$: $$\frac{h - 1}{9 - h} = 2.5$$ Multiply both sides: $$h - 1 = 2.5(9 - h)$$ $$h - 1 = 22.5 - 2.5h$$ $$h + 2.5h = 22.5 + 1$$ $$3.5h = 23.5$$ $$h = \frac{23.5}{3.5} = 6.714$$ 7. **Find $a$:** $$a|1 - h| = 2500$$ $$a(6.714 - 1) = 2500$$ $$a(5.714) = 2500$$ $$a = \frac{2500}{5.714} \approx 437.5$$ 8. **Write the function:** $$B(t) = 437.5|t - 6.714| - 900$$ 9. **Find days when $B(t) \geq 600$:** $$437.5|t - 6.714| - 900 \geq 600$$ Add 900: $$437.5|t - 6.714| \geq 1500$$ Divide by 437.5: $$|t - 6.714| \geq \frac{1500}{437.5} = 3.429$$ 10. **Solve inequality:** $$t - 6.714 \leq -3.429 \quad \text{or} \quad t - 6.714 \geq 3.429$$ $$t \leq 3.285 \quad \text{or} \quad t \geq 10.143$$ 11. **Interpret days:** - For $t \leq 3.285$, days 1, 2, 3 - For $t \geq 10.143$, days 11 through 30 (September has 30 days) 12. **Count days:** - Days 1 to 3: 3 days - Days 11 to 30: 20 days Total days with balance at least 600: $$3 + 20 = 23$$ **Final answer:** Kofi had at least 600 in his account for 23 days in September.