We need to solve the following equations for $x$: 28. $\frac{4}{x+1} = \frac{7}{3x-2}$ 29. $\frac{x+1}{2} + \frac{x-1}{3} = \frac{1}{6}$ 30. $\frac{1}{3}(x+2) = \frac{1}{5}(3x+2)$ 31. $\frac{1}{2}(x-1) - \frac{1}{6}(x+1) = 0$ 32. $\frac{1}{4}(x+5) - \frac{2x}{3} = 0$ 33. $\frac{4}{x} + 2 = 3$ 34. $\frac{6}{x} - 3 = 7$ 35. $\frac{9}{x} - 7 = 1$ 36. $-2 = 1 + \frac{3}{x}$ 37. $4 - \frac{4}{x} = 0$ 38. $5 - \frac{6}{x} = -1$ 39. $7 - \frac{3}{2x} = 1$ 40. $4 + \frac{5}{3x} = -1$ 41. $\frac{9}{2x} - 5 = 0$ 42. $\frac{x-1}{5} - \frac{x-1}{3} = 0$ 43. $\frac{x-1}{4} - \frac{2x-3}{5} = \frac{1}{20}$ 44. $\frac{4}{1-x} = \frac{3}{1+x}$ 45. $\frac{x+1}{4} - \frac{x}{3} = \frac{1}{12}$ 46. $\frac{2x+1}{8} - \frac{x-1}{3} = \frac{5}{24}$

1. Problem Description

We need to solve the following equations for $x$:

2

8. $\frac{4}{x+1} = \frac{7}{3x-2}$

2

9. $\frac{x+1}{2} + \frac{x-1}{3} = \frac{1}{6}$

3

0. $\frac{1}{3}(x+2) = \frac{1}{5}(3x+2)$

3

1. $\frac{1}{2}(x-1) - \frac{1}{6}(x+1) = 0$

3

2. $\frac{1}{4}(x+5) - \frac{2x}{3} = 0$

3

3. $\frac{4}{x} + 2 = 3$

3

4. $\frac{6}{x} - 3 = 7$

3

5. $\frac{9}{x} - 7 = 1$

3

6. $-2 = 1 + \frac{3}{x}$

3

7. $4 - \frac{4}{x} = 0$

3

8. $5 - \frac{6}{x} = -1$

3

9. $7 - \frac{3}{2x} = 1$

4

0. $4 + \frac{5}{3x} = -1$

4

1. $\frac{9}{2x} - 5 = 0$

4

2. $\frac{x-1}{5} - \frac{x-1}{3} = 0$

4

3. $\frac{x-1}{4} - \frac{2x-3}{5} = \frac{1}{20}$

4

4. $\frac{4}{1-x} = \frac{3}{1+x}$

4

5. $\frac{x+1}{4} - \frac{x}{3} = \frac{1}{12}$

4

6. $\frac{2x+1}{8} - \frac{x-1}{3} = \frac{5}{24}$

2. Solution Steps

2

8. $\frac{4}{x+1} = \frac{7}{3x-2}$

$4(3x-2) = 7(x+1)$

$12x - 8 = 7x + 7$

$5x = 15$

$x = 3$

2

9. $\frac{x+1}{2} + \frac{x-1}{3} = \frac{1}{6}$

Multiply by 6:

$3(x+1) + 2(x-1) = 1$

$3x+3 + 2x-2 = 1$

$5x+1 = 1$

$5x = 0$

$x = 0$

3

0. $\frac{1}{3}(x+2) = \frac{1}{5}(3x+2)$

Multiply by 15:

$5(x+2) = 3(3x+2)$

$5x+10 = 9x+6$

$4 = 4x$

$x = 1$

3

1. $\frac{1}{2}(x-1) - \frac{1}{6}(x+1) = 0$

Multiply by 6:

$3(x-1) - (x+1) = 0$

$3x-3 - x - 1 = 0$

$2x - 4 = 0$

$2x = 4$

$x = 2$

3

2. $\frac{1}{4}(x+5) - \frac{2x}{3} = 0$

Multiply by 12:

$3(x+5) - 8x = 0$

$3x+15-8x = 0$

$-5x + 15 = 0$

$5x = 15$

$x = 3$

3

3. $\frac{4}{x} + 2 = 3$

$\frac{4}{x} = 1$

$x = 4$

3

4. $\frac{6}{x} - 3 = 7$

$\frac{6}{x} = 10$

$6 = 10x$

$x = \frac{6}{10} = \frac{3}{5}$

3

5. $\frac{9}{x} - 7 = 1$

$\frac{9}{x} = 8$

$9 = 8x$

$x = \frac{9}{8}$

3

6. $-2 = 1 + \frac{3}{x}$

$-3 = \frac{3}{x}$

$-3x = 3$

$x = -1$

3

7. $4 - \frac{4}{x} = 0$

$4 = \frac{4}{x}$

$4x = 4$

$x = 1$

3

8. $5 - \frac{6}{x} = -1$

$6 = \frac{6}{x}$

$6x = 6$

$x = 1$

3

9. $7 - \frac{3}{2x} = 1$

$6 = \frac{3}{2x}$

$12x = 3$

$x = \frac{3}{12} = \frac{1}{4}$

4

0. $4 + \frac{5}{3x} = -1$

$\frac{5}{3x} = -5$

$5 = -15x$

$x = -\frac{5}{15} = -\frac{1}{3}$

4

1. $\frac{9}{2x} - 5 = 0$

$\frac{9}{2x} = 5$

$9 = 10x$

$x = \frac{9}{10}$

4

2. $\frac{x-1}{5} - \frac{x-1}{3} = 0$

$\frac{x-1}{5} = \frac{x-1}{3}$

$3(x-1) = 5(x-1)$

$3x-3 = 5x-5$

$2 = 2x$

$x = 1$

4

3. $\frac{x-1}{4} - \frac{2x-3}{5} = \frac{1}{20}$

Multiply by 20:

$5(x-1) - 4(2x-3) = 1$

$5x-5 - 8x + 12 = 1$

$-3x + 7 = 1$

$-3x = -6$

$x = 2$

4

4. $\frac{4}{1-x} = \frac{3}{1+x}$

$4(1+x) = 3(1-x)$

$4 + 4x = 3 - 3x$

$7x = -1$

$x = -\frac{1}{7}$

4

5. $\frac{x+1}{4} - \frac{x}{3} = \frac{1}{12}$

Multiply by 12:

$3(x+1) - 4x = 1$

$3x+3-4x = 1$

$-x + 3 = 1$

$-x = -2$

$x = 2$

4

6. $\frac{2x+1}{8} - \frac{x-1}{3} = \frac{5}{24}$

Multiply by 24:

$3(2x+1) - 8(x-1) = 5$

$6x+3 - 8x + 8 = 5$

$-2x + 11 = 5$

$-2x = -6$

$x = 3$

3. Final Answer

2

8. $x = 3$

2

9. $x = 0$

3

0. $x = 1$

3

1. $x = 2$

3

2. $x = 3$

3

3. $x = 4$

3

4. $x = \frac{3}{5}$

3

5. $x = \frac{9}{8}$

3

6. $x = -1$

3

7. $x = 1$

3

8. $x = 1$

3

9. $x = \frac{1}{4}$

4

0. $x = -\frac{1}{3}$

4

1. $x = \frac{9}{10}$

4

2. $x = 1$

4

3. $x = 2$

4

4. $x = -\frac{1}{7}$

4

5. $x = 2$

4

6. $x = 3$