We are given the equation:

$\frac{2x - 3}{9} + \frac{4x - 2}{7} - 2 = \frac{6x - 2}{4}$

To solve this equation, let's go step by step:

Step 1: Eliminate the fractions

To eliminate the fractions, multiply every term by the least common denominator (LCD) of the denominators $9$, $7$, and $4$. The LCD of 9, 7, and 4 is 252.

Multiply each term by 252:

$252 \times \left(\frac{2x - 3}{9}\right) + 252 \times \left(\frac{4x - 2}{7}\right) - 252 \times 2 = 252 \times \left(\frac{6x - 2}{4}\right)$

This simplifies to:

$28(2x - 3) + 36(4x - 2) - 504 = 63(6x - 2)$

Step 2: Expand the terms

Now, expand each term:

$28(2x - 3) = 56x - 84$ $36(4x - 2) = 144x - 72$ $63(6x - 2) = 378x - 126$

Substituting these into the equation gives:

$56x - 84 + 144x - 72 - 504 = 378x - 126$

Step 3: Combine like terms

On the left-hand side:

$56x + 144x - 84 - 72 - 504 = 200x - 660$

So the equation becomes:

$200x - 660 = 378x - 126$

Step 4: Solve for $x$

To isolate $x$, subtract $200x$ from both sides:

$-660 = 178x - 126$

Next, add 126 to both sides:

$-534 = 178x$

Finally, divide by 178:

$x = \frac{-534}{178} = -3$

Final Answer:

$x = -3$

Would you like a detailed breakdown of any of these steps? Here are some related questions:

1. How can you verify the solution by substituting $x = -3$ back into the original equation?
2. What strategies can you use to solve equations with fractions?
3. How is the least common denominator (LCD) determined for multiple fractions?
4. How do you handle equations with variables on both sides?
5. What are common mistakes to avoid when working with fractions in algebra?

Tip: Always double-check your solution by substituting it back into the original equation to ensure it satisfies all terms.