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Let's go through these equations one by one.

### 1. \( \frac{3x}{4} - 6 = \frac{x}{4} \)
- **Step 1**: Eliminate the fractions by multiplying the entire equation by 4:
  \[
  3x - 24 = x
  \]
- **Step 2**: Move all terms involving \(x\) to one side:
  \[
  3x - x = 24
  \]
- **Step 3**: Simplify:
  \[
  2x = 24
  \]
- **Step 4**: Solve for \(x\):
  \[
  x = 12
  \]

### 2. \( x - \frac{8}{3} - \frac{1}{x} = 0 \)
- **Step 1**: Move \(\frac{8}{3}\) and \(\frac{1}{x}\) to the other side:
  \[
  x = \frac{8}{3} + \frac{1}{x}
  \]
- **Step 2**: Multiply through by \(x\) to eliminate the fraction:
  \[
  x^2 = \frac{8}{3}x + 1
  \]
- **Step 3**: Rearrange to form a quadratic equation:
  \[
  3x^2 - 8x - 3 = 0
  \]
Now you can solve this quadratic equation using the quadratic formula.

### 3. \( x + 2 = \frac{9}{x + 2} \)
- **Step 1**: Multiply both sides by \(x + 2\):
  \[
  (x + 2)^2 = 9
  \]
- **Step 2**: Take the square root of both sides:
  \[
  x + 2 = \pm 3
  \]
- **Step 3**: Solve for \(x\):
  - If \(x + 2 = 3\), then \(x = 1\)
  - If \(x + 2 = -3\), then \(x = -5\)

### 4. \( x + 2 = \frac{5x}{x - 4} - \frac{14}{x - 4} \)
- **Step 1**: Combine the right-hand side:
  \[
  x + 2 = \frac{5x - 14}{x - 4}
  \]
- **Step 2**: Multiply both sides by \(x - 4\) to eliminate the fraction:
  \[
  (x + 2)(x - 4) = 5x - 14
  \]
Now expand both sides and simplify to form a quadratic equation to solve for \(x\).

### 5. \( 1 - \frac{x - 4}{1 - x} = \frac{12}{x - 1} \)
- **Step 1**: Simplify \(1 - \frac{x - 4}{1 - x}\) using properties of negative signs:
  \[
  1 - \frac{x - 4}{1 - x} = 1 + 1 = 2
  \]
So the equation becomes:
  \[
  2 = \frac{12}{x - 1}
  \]
- **Step 2**: Multiply both sides by \(x - 1\):
  \[
  2(x - 1) = 12
  \]
- **Step 3**: Solve for \(x\):
  \[
  2x - 2 = 12 \quad \Rightarrow \quad 2x = 14 \quad \Rightarrow \quad x = 7
  \]

### 6. \( \frac{3x}{2x - 5} - \frac{7}{3x + 1} = \frac{11}{4} \)
For this equation, you need to find a common denominator on the left-hand side, then solve as a rational equation.

### 7. \( \frac{x + 2}{x - 2} - \frac{x - 2}{x + 2} = \frac{7}{4} \)
This can be solved by first combining the fractions on the left side.

Solve for the value/s of x in the given equations:
1. 3x/4 - 6 = x/4
2. x - 8/3 - 1/x = 0
3. x + 2 = 9/(x + 2)
4. x + 2 = 5x/(x - 4) - 14/(x - 4)
5. 1 - (x - 4)/(1 - x) = 12/(x - 1)
6. 3x/(2x - 5) - 7/(3x + 1) = 11/4
7. (x + 2)/(x - 2) - (x - 2)/(x + 2) = 7/4

This page provides step-by-step solutions to a variety of algebraic equations, including linear, quadratic, and rational expressions. Suitable for students in grades 8-10, these problems explore solving for x using algebraic methods like cross-multiplication and factoring.

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