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The equation you've uploaded is:

\[
\frac{2K - 3}{x - 1} + \frac{3Kx - 2}{x + 1} = 2K + 3
\]

Let's solve this step-by-step for \( K \) (assuming \( K \) is the variable we are solving for).

### Step 1: Eliminate the denominators
We will multiply both sides by the least common denominator (LCD) of the two fractions, which is \( (x - 1)(x + 1) \).

The equation becomes:
\[
(2K - 3)(x + 1) + (3Kx - 2)(x - 1) = (2K + 3)(x - 1)(x + 1)
\]

### Step 2: Expand both sides

#### Left-hand side:
1. Expand \( (2K - 3)(x + 1) \):
   \[
   (2K - 3)(x + 1) = 2Kx + 2K - 3x - 3
   \]

2. Expand \( (3Kx - 2)(x - 1) \):
   \[
   (3Kx - 2)(x - 1) = 3Kx^2 - 3Kx - 2x + 2
   \]

The left-hand side becomes:
\[
2Kx + 2K - 3x - 3 + 3Kx^2 - 3Kx - 2x + 2
\]
Simplifying:
\[
3Kx^2 + (2Kx - 3Kx - 3x - 2x) + (2K - 3 + 2) = 3Kx^2 - x(4 + K) + 2K - 1
\]

#### Right-hand side:
1. The right-hand side \( (2K + 3)(x^2 - 1) \):
   \[
   (2K + 3)(x^2 - 1) = (2K + 3)x^2 - (2K + 3)
   \]

### Step 3: Combine both sides
Now set both expressions equal:
\[
3Kx^2 - x(4 + K) + (2K - 1) = (2K + 3)x^2 - (2K + 3)
\]

### Step 4: Compare coefficients
Equate the coefficients of \( x^2 \), \( x \), and the constant terms.

For \( x^2 \):
\[
3K = 2K + 3
\]
Solving for \( K \):
\[
3K - 2K = 3 \implies K = 3
\]

### Step 5: Substitute \( K = 3 \) into the equation
Now, we can verify by substituting \( K = 3 \) into the original equation to check if both sides are equal.

Would you like further clarification on any part of the solution? Here are five follow-up questions:

1. How do you find the least common denominator of rational expressions?
2. What happens when you multiply both sides of an equation by a common denominator?
3. Why is expanding and simplifying important in solving equations like this one?
4. How do you equate the coefficients of like terms in an algebraic expression?
5. What other methods can be used to solve similar equations?

**Tip:** When solving rational equations, always check for extraneous solutions caused by multiplying by terms that could potentially be zero.

Solve the equation: (2K - 3)/(x - 1) + (3Kx - 2)/(x + 1) = 2K + 3

This problem involves solving a rational equation with variables in the denominator. Key concepts include multiplying by a common denominator, expanding terms, and equating coefficients. Follow the detailed steps to solve for K.

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