1. Problem 7: Simplify $m + n + \frac{1}{m - n}$. We are given an expression and asked to simplify it or write it as a single fraction. 2. To combine terms into a single fraction, find a common denominator. Here, the denominator is $m - n$. 3. Rewrite $m + n$ as $\frac{(m + n)(m - n)}{m - n}$ to have the same denominator: $$m + n = \frac{(m + n)(m - n)}{m - n}$$ 4. So the entire expression becomes: $$\frac{(m + n)(m - n)}{m - n} + \frac{1}{m - n} = \frac{(m + n)(m - n) + 1}{m - n}$$ 5. Expand the numerator: $$(m + n)(m - n) = m^2 - n^2$$ 6. Substitute back: $$\frac{m^2 - n^2 + 1}{m - n}$$ 7. This matches option C. --- 8. Problem 8: Simplify $\frac{3 - x}{x - 4} \div \frac{2x - 6}{5x - 20}$. 9. Division of fractions means multiply by the reciprocal: $$\frac{3 - x}{x - 4} \times \frac{5x - 20}{2x - 6}$$ 10. Factor where possible: $$2x - 6 = 2(x - 3)$$ $$5x - 20 = 5(x - 4)$$ 11. Substitute factors: $$\frac{3 - x}{x - 4} \times \frac{5(x - 4)}{2(x - 3)}$$ 12. Cancel common factor $x - 4$: $$\frac{3 - x}{\cancel{x - 4}} \times \frac{5\cancel{(x - 4)}}{2(x - 3)} = \frac{3 - x}{1} \times \frac{5}{2(x - 3)}$$ 13. Note $3 - x = -(x - 3)$, so: $$\frac{3 - x}{1} = - (x - 3)$$ 14. Substitute: $$- (x - 3) \times \frac{5}{2(x - 3)}$$ 15. Cancel $x - 3$: $$- \cancel{(x - 3)} \times \frac{5}{2 \cancel{(x - 3)}} = - \frac{5}{2}$$ 16. Final answer is $-\frac{5}{2}$, which is option D. --- 17. Problem 9: Simplify $\frac{\sqrt{8}}{\sqrt{75}} + \frac{7}{\sqrt{98}}$. 18. Simplify radicals: $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$ $$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$$ $$\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}$$ 19. Substitute: $$\frac{2\sqrt{2}}{5\sqrt{3}} + \frac{7}{7\sqrt{2}} = \frac{2\sqrt{2}}{5\sqrt{3}} + \frac{1}{\sqrt{2}}$$ 20. Rationalize denominators: $$\frac{2\sqrt{2}}{5\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{6}}{15}$$ $$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$ 21. Find common denominator 30: $$\frac{2\sqrt{6}}{15} = \frac{4\sqrt{6}}{30}$$ $$\frac{\sqrt{2}}{2} = \frac{15\sqrt{2}}{30}$$ 22. Add: $$\frac{4\sqrt{6} + 15\sqrt{2}}{30}$$ 23. This matches option B. --- 24. Problem 10: Find $n$ if $2\sqrt{180} - \frac{\sqrt{80}}{2} = \sqrt{n}$. 25. Simplify radicals: $$\sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}$$ $$\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$$ 26. Substitute: $$2 \times 6\sqrt{5} - \frac{4\sqrt{5}}{2} = 12\sqrt{5} - 2\sqrt{5} = 10\sqrt{5}$$ 27. We have: $$10\sqrt{5} = \sqrt{n}$$ 28. Square both sides: $$\left(10\sqrt{5}\right)^2 = n$$ $$100 \times 5 = n$$ $$n = 500$$ 29. Final answer is 500, option A.