\( \begin{array}{ll}\text { i) } 5 x-9=3 x & \text { ii) } 3(7 x+8)=5(3 x+1) \\ \text { iii) } \frac{3 x+2}{4}=\frac{2 x+6}{5} & \text { iv) } \frac{2 x+7}{4}-\frac{x+1}{6}=\frac{5-x}{3} \\ \text { v) } 7(x+5)=5(5-x) & \text { vi) } 6 x-9=3 x \\ \text { vii) } 3(4-5 x)-2(x+4)=4 & \text { viii) } \frac{x+1}{x}=\frac{4}{x^{2}-x}=\frac{x+1}{x-1} \\ \text { ix) } 4(x-7)=3(2 x+1)-5 & \text { x) } \frac{3 x-1}{x-2}=\frac{9 x+5}{3 x+5} \\ \text { ix } 5 x-4+3(2 x-5)=8 x-25 & \text { xii) } \frac{3 x-4}{6}=\frac{5+2 x}{3}=0\end{array} \)

**i)** We start with \[ 5x - 9 = 3x. \] Subtract \(3x\) from both sides: \[ 5x - 3x - 9 = 0 \quad \Longrightarrow \quad 2x - 9 = 0. \] Add 9 to both sides: \[ 2x = 9. \] Divide by 2: \[ x = \frac{9}{2}. \] --- **ii)** The equation is \[ 3(7x + 8) = 5(3x + 1). \] Expand both sides: \[ 21x + 24 = 15x + 5. \] Subtract \(15x\) from both sides: \[ 6x + 24 = 5. \] Subtract 24: \[ 6x = -19. \] Divide by 6: \[ x = -\frac{19}{6}. \] --- **iii)** We have \[ \frac{3x+2}{4} = \frac{2x+6}{5}. \] Multiply both sides by 20 (the least common multiple of 4 and 5): \[ 20\cdot\frac{3x+2}{4} = 20\cdot\frac{2x+6}{5} \quad \Longrightarrow \quad 5(3x+2) = 4(2x+6). \] This gives: \[ 15x + 10 = 8x + 24. \] Subtract \(8x\): \[ 7x + 10 = 24. \] Subtract 10: \[ 7x = 14. \] Divide by 7: \[ x = 2. \] --- **iv)** The equation is \[ \frac{2x+7}{4} - \frac{x+1}{6} = \frac{5-x}{3}. \] Multiply both sides by 12 (the least common multiple of 4, 6, and 3): \[ 12\cdot\frac{2x+7}{4} - 12\cdot\frac{x+1}{6} = 12\cdot\frac{5-x}{3}. \] Simplify each term: \[ 3(2x+7) - 2(x+1) = 4(5-x). \] Compute the left-hand side: \[ 6x + 21 - 2x - 2 = 4x + 19. \] And the right-hand side: \[ 20 - 4x. \] Thus, we have: \[ 4x + 19 = 20 - 4x. \] Add \(4x\) to both sides: \[ 8x + 19 = 20. \] Subtract 19: \[ 8x = 1. \] Divide by 8: \[ x = \frac{1}{8}. \] --- **v)** The equation is \[ 7(x+5) = 5(5-x). \] Expand both sides: \[ 7x + 35 = 25 - 5x. \] Add \(5x\) to both sides: \[ 12x + 35 = 25. \] Subtract 35: \[ 12x = -10. \] Divide by 12: \[ x = -\frac{10}{12} = -\frac{5}{6}. \] --- **vi)** We have \[ 6x - 9 = 3x. \] Subtract \(3x\): \[ 3x - 9 = 0. \] Add 9: \[ 3x = 9. \] Divide by 3: \[ x = 3. \] --- **vii)** The equation is \[ 3(4-5x) - 2(x+4) = 4. \] Expand: \[ 12 - 15x - 2x - 8 = 4. \] Combine like terms: \[ (12 - 8) - 17x = 4 \quad \Longrightarrow \quad 4 - 17x = 4. \]