We are given the equation:

\[
\frac{9}{x} + \frac{3}{12x - 4} = \frac{2}{12x - 4}
\]

### Steps to solve this equation:

1. First, observe that both sides of the equation contain terms with \(12x - 4\) in the denominator. To simplify things, let's introduce a substitution for \(12x - 4\):

   \[
   y = 12x - 4
   \]
   
   So the equation becomes:

   \[
   \frac{9}{x} + \frac{3}{y} = \frac{2}{y}
   \]

2. Now, subtract \(\frac{2}{y}\) from both sides:

   \[
   \frac{9}{x} + \frac{3}{y} - \frac{2}{y} = 0
   \]

3. Combine the terms with \(y\):

   \[
   \frac{9}{x} + \frac{1}{y} = 0
   \]

4. Next, solve for \(\frac{9}{x}\):

   \[
   \frac{9}{x} = -\frac{1}{y}
   \]

5. Cross-multiply to eliminate the fractions:

   \[
   9y = -x
   \]

6. Solve for \(x\):

   \[
   x = -9y
   \]

7. Substitute back \(y = 12x - 4\):

   \[
   x = -9(12x - 4)
   \]

8. Expand the right side:

   \[
   x = -108x + 36
   \]

9. Add \(108x\) to both sides:

   \[
   109x = 36
   \]

10. Solve for \(x\):

   \[
   x = \frac{36}{109}
   \]

Thus, the solution is:

\[
x = \frac{36}{109}
\]

Would you like more details or have any questions?

### Related questions:
1. How can you simplify complex rational expressions?
2. What is substitution in algebra, and when should it be used?
3. How can you solve equations with variables in the denominator?
4. What is the significance of cross-multiplying when solving fractions?
5. How do you solve linear equations with more than one variable?

**Tip:** When solving equations with fractions, it’s often helpful to find a common denominator or use substitution to simplify the problem.

This page covers the step-by-step solution to the rational equation 9/x + 3/(12x - 4) = 2/(12x - 4). Learn how to use substitution and cross-multiplication to simplify and solve equations with variables in the denominator. Suitable for students in Grades 8-10.

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