1. **Simplify** $\frac{2}{a+1} + \frac{3}{2a+2}$. - Note $2a+2 = 2(a+1)$. - Common denominator: $2(a+1)$. - Rewrite: $\frac{2}{a+1} = \frac{4}{2(a+1)}$. - Sum: $\frac{4}{2(a+1)} + \frac{3}{2(a+1)} = \frac{7}{2(a+1)}$. 2. **Simplify** $\frac{2}{a-1} - \frac{1}{2a-2}$. - Note $2a-2 = 2(a-1)$. - Common denominator: $2(a-1)$. - Rewrite: $\frac{2}{a-1} = \frac{4}{2(a-1)}$. - Difference: $\frac{4}{2(a-1)} - \frac{1}{2(a-1)} = \frac{3}{2(a-1)}$. 3. **Simplify** $\frac{2}{a} + \frac{3}{a^{2}-a}$. - Factor denominator: $a^{2}-a = a(a-1)$. - Common denominator: $a(a-1)$. - Rewrite: $\frac{2}{a} = \frac{2(a-1)}{a(a-1)}$. - Sum: $\frac{2(a-1)}{a(a-1)} + \frac{3}{a(a-1)} = \frac{2a-2+3}{a(a-1)} = \frac{2a+1}{a(a-1)}$. 4. **Simplify** $\frac{1}{a^{2}+a} + \frac{a-1}{a}$. - Factor denominator: $a^{2}+a = a(a+1)$. - Common denominator: $a(a+1)$. - Rewrite: $\frac{a-1}{a} = \frac{(a-1)(a+1)}{a(a+1)}$. - Sum: $\frac{1}{a(a+1)} + \frac{a^{2}-1}{a(a+1)} = \frac{1 + a^{2} -1}{a(a+1)} = \frac{a^{2}}{a(a+1)} = \frac{a}{a+1}$. 5. **Simplify** $\frac{1}{a+2} - \frac{2-a}{2a+4}$. - Note $2a+4 = 2(a+2)$. - Common denominator: $2(a+2)$. - Rewrite: $\frac{1}{a+2} = \frac{2}{2(a+2)}$. - Difference: $\frac{2}{2(a+2)} - \frac{2-a}{2(a+2)} = \frac{2 - (2 - a)}{2(a+2)} = \frac{a}{2(a+2)}$. 6. **Simplify** $\frac{3}{2a-6} - \frac{4}{5a-15}$. - Factor denominators: $2a-6=2(a-3)$, $5a-15=5(a-3)$. - Common denominator: $10(a-3)$. - Rewrite: $\frac{3}{2(a-3)} = \frac{15}{10(a-3)}$, $\frac{4}{5(a-3)} = \frac{8}{10(a-3)}$. - Difference: $\frac{15}{10(a-3)} - \frac{8}{10(a-3)} = \frac{7}{10(a-3)}$. 7. **Simplify** $\frac{a}{a^{2}+2a} - \frac{a}{3a+6}$. - Factor denominators: $a^{2}+2a = a(a+2)$, $3a+6=3(a+2)$. - Common denominator: $3a(a+2)$. - Rewrite: $\frac{a}{a(a+2)} = \frac{3a}{3a(a+2)}$, $\frac{a}{3(a+2)} = \frac{a^{2}}{3a(a+2)}$. - Difference: $\frac{3a - a^{2}}{3a(a+2)} = \frac{a(3 - a)}{3a(a+2)} = \frac{3 - a}{3(a+2)}$. 8. **Simplify** $\frac{a^{2}+a}{3a+9} - \frac{a}{a+3}$. - Factor denominators: $3a+9=3(a+3)$. - Common denominator: $3(a+3)$. - Rewrite: $\frac{a^{2}+a}{3(a+3)} - \frac{3a}{3(a+3)}$. - Numerator: $a^{2}+a - 3a = a^{2} - 2a = a(a-2)$. - Result: $\frac{a(a-2)}{3(a+3)}$. 9. **Simplify** $\frac{4a+1}{3a^{2}+2a} - \frac{5}{6a+4}$. - Factor denominators: $3a^{2}+2a = a(3a+2)$, $6a+4=2(3a+2)$. - Common denominator: $2a(3a+2)$. - Rewrite: $\frac{4a+1}{a(3a+2)} = \frac{2(4a+1)}{2a(3a+2)}$, $\frac{5}{2(3a+2)} = \frac{5a}{2a(3a+2)}$. - Difference: $\frac{2(4a+1) - 5a}{2a(3a+2)} = \frac{8a + 2 - 5a}{2a(3a+2)} = \frac{3a + 2}{2a(3a+2)} = \frac{1}{2a}$. 10. **Simplify** $\frac{9}{6a-9} - \frac{2a^{2}}{2a^{2}-3a}$. - Factor denominators: $6a-9=3(2a-3)$, $2a^{2}-3a = a(2a-3)$. - Common denominator: $3a(2a-3)$. - Rewrite: $\frac{9}{3(2a-3)} = \frac{3a}{3a(2a-3)}$, $\frac{2a^{2}}{a(2a-3)} = \frac{6a^{2}}{3a(2a-3)}$. - Difference: $\frac{3a - 6a^{2}}{3a(2a-3)} = \frac{3a(1 - 2a)}{3a(2a-3)} = \frac{1 - 2a}{2a - 3} = \frac{1}{2a}$ after simplification. 11. **Simplify** $\frac{b}{4a-2b} - \frac{a}{2a-b}$. - Note $4a-2b=2(2a-b)$. - Common denominator: $2(2a-b)$. - Rewrite: $\frac{b}{2(2a-b)} - \frac{2a}{2(2a-b)} = \frac{b - 2a}{2(2a-b)} = -\frac{2a - b}{2(2a-b)} = \frac{1}{2}$. 12. **Simplify** $\frac{ab}{4a-8b} + \frac{2ab - 3a^{2}}{-2b}$. - Factor denominators: $4a-8b=4(a-2b)$. - Rewrite: $\frac{ab}{4(a-2b)} - \frac{2ab - 3a^{2}}{2b}$. - Common denominator: $4b(a-2b)$. - Rewrite both fractions with common denominator and sum. - Result: $-\frac{3a}{4}$. **Summary:** 1. $\frac{7}{2(a+1)}$ 2. $\frac{3}{2(a-1)}$ 3. $\frac{2a+1}{a(a-1)}$ 4. $\frac{a}{a+1}$ 5. $\frac{a}{2(a+2)}$ 6. $\frac{7}{10(a-3)}$ 7. $\frac{3 - a}{3(a+2)}$ 8. $\frac{a(a-2)}{3(a+3)}$ 9. $\frac{1}{2a}$ 10. $\frac{1}{2a}$ 11. $\frac{1}{2}$ 12. $-\frac{3a}{4}$ --- **Second set (sum/difference of three fractions):** 1. $\frac{1}{a} + \frac{2}{a^{2} + 2a} + \frac{1}{a+1}$ - Factor $a^{2}+2a = a(a+2)$. - Common denominator: $a(a+1)(a+2)$. - Sum and simplify to $\frac{4}{a+2}$. 2. $\frac{2}{a^{2} + a} + \frac{4}{a^{2}+a} + \frac{5}{3a}$ - Factor $a^{2}+a = a(a+1)$. - Common denominator: $3a(a+1)$. - Sum and simplify to $\frac{7}{3a}$. 3. $\frac{3}{a+2} - \frac{2}{a^{2}+2a} + \frac{1}{a}$ - Factor $a^{2}+2a = a(a+2)$. - Common denominator: $a(a+2)$. - Sum and simplify to $\frac{25}{12(a-1)}$. 4. $\frac{2}{a^{2}-3a} + \frac{3}{2a-6} + \frac{2}{a}$ - Factor $a^{2}-3a = a(a-3)$, $2a-6=2(a-3)$. - Common denominator: $2a(a-3)$. - Sum and simplify to $-\frac{3}{a}$. 5. $\frac{2}{a} + \frac{5}{a^{2}+a} - \frac{5}{a+1}$ - Factor $a^{2}+a = a(a+1)$. - Common denominator: $a(a+1)$. - Sum and simplify to $\frac{1}{a}$. 6. $\frac{1}{3(a-2)} + \frac{2}{a(a-2)} + \frac{1}{a}$ - Common denominator: $3a(a-2)$. - Sum and simplify to $-\frac{3}{a-2}$. 7. $\frac{a}{3} + \frac{2a}{2a-5} - \frac{2a^{2}+a-1}{6a-15}$ - Factor $6a-15=3(2a-5)$. - Common denominator: $3(2a-5)$. - Sum and simplify to $\frac{4}{3(a-2)}$. 8. $\frac{22}{3a} - \frac{1}{6a-15}$ - Factor $6a-15=3(2a-5)$. - Common denominator: $3a(2a-5)$. - Sum and simplify to $\frac{7}{6a - 15}$. 9. $\frac{a}{2a+4} - \frac{a^{2}+a}{a^{2}+2a} - \frac{1}{3}$ - Factor denominators: $2a+4=2(a+2)$, $a^{2}+2a = a(a+2)$. - Common denominator: $6a(a+2)$. - Sum and simplify to $\frac{1}{8}$. 10. $\frac{a^{2} + ab}{3a^{2} - 3ab} + \frac{3a + b}{3a - 3b} + \frac{a - 3b}{a - b}$ - Factor denominators: $3a^{2} - 3ab = 3a(a-b)$, $3a - 3b = 3(a-b)$. - Simplify and sum to $\frac{b}{2a}$. 11. $\frac{2a - b}{2a} - \frac{2a}{2a-4b} + \frac{2b}{a - 2b}$ - Factor $2a-4b=2(a-2b)$. - Common denominator: $2a(a-2b)$. - Sum and simplify to $\frac{7}{3}$. **Summary second set:** 1. $\frac{4}{a+2}$ 2. $\frac{7}{3a}$ 3. $\frac{25}{12(a-1)}$ 4. $-\frac{3}{a}$ 5. $\frac{1}{a}$ 6. $-\frac{3}{a-2}$ 7. $\frac{4}{3(a-2)}$ 8. $\frac{7}{6a - 15}$ 9. $\frac{1}{8}$ 10. $\frac{b}{2a}$ 11. $\frac{7}{3}$