1. Problem 16: Given the function $g(x) = -f(x-3)$, solve the inequality $g(x) < 0$ and find the interval where this holds. 2. Since $g(x) = -f(x-3)$, the inequality $g(x) < 0$ is equivalent to: $$-f(x-3) < 0$$ 3. Multiply both sides by $-1$ (remember to reverse the inequality sign): $$f(x-3) > 0$$ 4. To find where $g(x) < 0$, we need to find where $f(x-3) > 0$. 5. Let $t = x - 3$. Then the inequality becomes: $$f(t) > 0$$ 6. From the graph of $f(x)$, identify the intervals where $f(t) > 0$. 7. Suppose from the graph $f(t) > 0$ for $t$ in the interval $(a, b)$. 8. Substitute back $t = x - 3$: $$x - 3 \\in (a, b) \\Rightarrow x \\in (a + 3, b + 3)$$ 9. Therefore, the solution to $g(x) < 0$ is: $$x \\in (a + 3, b + 3)$$ --- 1. Problem 17: Antanas and Balys working together can complete a task 16 hours faster than Antanas alone, and 9 hours faster than Balys alone. Find how many hours Balys alone would take to complete the task. 2. Let $A$ be the time Antanas takes alone, and $B$ be the time Balys takes alone. 3. Working together, their time is $T$. 4. Given: $$T = A - 16$$ $$T = B - 9$$ 5. Their combined work rate is: $$\frac{1}{A} + \frac{1}{B} = \frac{1}{T}$$ 6. Substitute $T$ from above: $$\frac{1}{A} + \frac{1}{B} = \frac{1}{A - 16}$$ and $$\frac{1}{A} + \frac{1}{B} = \frac{1}{B - 9}$$ 7. From the equalities of $T$, we have: $$A - 16 = B - 9 \\Rightarrow A - B = 7$$ 8. Express $A$ as: $$A = B + 7$$ 9. Substitute $A$ and $T$ into the combined rate equation: $$\frac{1}{B + 7} + \frac{1}{B} = \frac{1}{B - 9}$$ 10. Multiply both sides by $(B + 7) B (B - 9)$ to clear denominators: $$B (B - 9) + (B + 7)(B - 9) = (B + 7) B$$ 11. Expand terms: $$B^2 - 9B + (B^2 - 2B - 63) = B^2 + 7B$$ 12. Combine like terms: $$B^2 - 9B + B^2 - 2B - 63 = B^2 + 7B$$ $$2B^2 - 11B - 63 = B^2 + 7B$$ 13. Bring all terms to one side: $$2B^2 - 11B - 63 - B^2 - 7B = 0$$ $$B^2 - 18B - 63 = 0$$ 14. Solve quadratic equation: $$B = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot (-63)}}{2} = \frac{18 \pm \sqrt{324 + 252}}{2} = \frac{18 \pm \sqrt{576}}{2} = \frac{18 \pm 24}{2}$$ 15. Two solutions: $$B = \frac{18 + 24}{2} = 21$$ $$B = \frac{18 - 24}{2} = -3$$ 16. Time cannot be negative, so $B = 21$ hours. **Final answers:** - For problem 16: $g(x) < 0$ when $x$ is in the interval shifted by 3 to the right from where $f(x) > 0$. - For problem 17: Balys alone would take 21 hours to complete the task.