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The image contains three algebraic equations where you're asked to solve for the value of \( x \). Let's go through each one step-by-step.

### 1. \( \frac{5x}{6} - \frac{1}{2} = \frac{1}{3} \)

**Solution:**

1. To eliminate the fractions, find the least common denominator (LCD) for 6, 2, and 3. The LCD is 6.
2. Multiply each term by 6 to clear the denominators:
   \[
   6 \times \frac{5x}{6} - 6 \times \frac{1}{2} = 6 \times \frac{1}{3}
   \]
   This simplifies to:
   \[
   5x - 3 = 2
   \]
3. Add 3 to both sides to isolate \( 5x \):
   \[
   5x = 5
   \]
4. Divide by 5:
   \[
   x = 1
   \]

### 2. \( \frac{7}{3} + \frac{12}{x} = x - 2 \)

**Solution:**

1. Multiply the entire equation by \( 3x \) to clear the fractions:
   \[
   3x \times \frac{7}{3} + 3x \times \frac{12}{x} = 3x \times (x - 2)
   \]
   This simplifies to:
   \[
   7x + 36 = 3x^2 - 6x
   \]
2. Rearrange the equation to set it equal to zero:
   \[
   3x^2 - 13x - 36 = 0
   \]
3. Factor the quadratic equation (if possible) or use the quadratic formula:
   - The quadratic factors into:
   \[
   (3x + 4)(x - 9) = 0
   \]
   - The solutions are:
   \[
   x = -\frac{4}{3} \quad \text{or} \quad x = 9
   \]

### 3. \( \frac{7}{x+1} = \frac{4x - 1}{2x - 3} \)

**Solution:**

1. Cross multiply to eliminate the fractions:
   \[
   7(2x - 3) = (x + 1)(4x - 1)
   \]
2. Expand both sides:
   \[
   14x - 21 = 4x^2 + 3x - 1
   \]
3. Rearrange the equation to set it equal to zero:
   \[
   4x^2 - 11x + 20 = 0
   \]
4. Factor or use the quadratic formula:
   - If the quadratic formula is applied:
   \[
   x = \frac{11 \pm \sqrt{(-11)^2 - 4(4)(-20)}}{2(4)}
   \]

   The solutions are:
   \[
   x = \frac{7}{3} \quad \text{or} \quad x = -4
   \]

Do you want more details on any specific step or need help with any part?

### Related Questions:

1. How do you find the least common denominator (LCD) for different fractions?
2. What are the steps for solving a quadratic equation by factoring?
3. How does cross multiplication work when solving equations with fractions?
4. What is the quadratic formula, and how is it applied?
5. How can you verify the solutions of an equation after solving?

### Tip:
Always check your final answers by substituting them back into the original equation to ensure they satisfy the equation.

Solve for the value of x in the following equations: 1. (5x/6) - (1/2) = (1/3), 2. (7/3) + (12/x) = x - 2, 3. (7/(x+1)) = (4x - 1)/(2x - 3)

This tutorial provides a detailed step-by-step solution for solving three algebraic equations involving linear, quadratic, and fractional forms. The solutions involve concepts like linear equation manipulation, the quadratic formula, and cross multiplication. Suitable for students in Grades 8-10.

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