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.