To solve the equation \(\frac{2}{x-3} - \frac{3}{x-2} = \frac{2}{3} - \frac{3}{2}\), follow these steps:

### Step 1: Simplify the right-hand side
Calculate \(\frac{2}{3} - \frac{3}{2}\) by finding a common denominator:

\[
\frac{2}{3} - \frac{3}{2} = \frac{2 \times 2}{3 \times 2} - \frac{3 \times 3}{2 \times 3} = \frac{4}{6} - \frac{9}{6} = \frac{4 - 9}{6} = \frac{-5}{6}
\]

So, the equation becomes:

\[
\frac{2}{x-3} - \frac{3}{x-2} = \frac{-5}{6}
\]

### Step 2: Combine the left-hand side
To combine the fractions on the left-hand side, find a common denominator:

\[
\frac{2}{x-3} - \frac{3}{x-2} = \frac{2(x-2) - 3(x-3)}{(x-3)(x-2)}
\]

Expand and simplify the numerator:

\[
2(x-2) = 2x - 4
\]
\[
3(x-3) = 3x - 9
\]

So the numerator becomes:

\[
2x - 4 - (3x - 9) = 2x - 4 - 3x + 9 = -x + 5
\]

The equation is now:

\[
\frac{-x + 5}{(x-3)(x-2)} = \frac{-5}{6}
\]

### Step 3: Cross-multiply
Cross-multiply to eliminate the fractions:

\[
-6(x - 3)(x - 2) = -5(-x + 5)
\]

### Step 4: Expand and solve the equation
First, expand both sides:

Left-hand side:
\[
-6(x^2 - 5x + 6) = -6x^2 + 30x - 36
\]

Right-hand side:
\[
-5(-x + 5) = 5x - 25
\]

The equation is now:

\[
-6x^2 + 30x - 36 = 5x - 25
\]

### Step 5: Rearrange the equation
Move all terms to one side of the equation:

\[
-6x^2 + 30x - 36 - 5x + 25 = 0
\]

Combine like terms:

\[
-6x^2 + 25x - 11 = 0
\]

### Step 6: Solve the quadratic equation
You can solve the quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \(a = -6\), \(b = 25\), and \(c = -11\).

Substitute these values into the quadratic formula:

\[
x = \frac{-25 \pm \sqrt{25^2 - 4(-6)(-11)}}{2(-6)}
\]
\[
x = \frac{-25 \pm \sqrt{625 - 264}}{-12}
\]
\[
x = \frac{-25 \pm \sqrt{361}}{-12}
\]
\[
x = \frac{-25 \pm 19}{-12}
\]

This gives two solutions:

\[
x = \frac{-25 + 19}{-12} = \frac{-6}{-12} = \frac{1}{2}
\]
\[
x = \frac{-25 - 19}{-12} = \frac{-44}{-12} = \frac{11}{3}
\]

### Final Answer
The solutions are \(x = \frac{1}{2}\) and \(x = \frac{11}{3}\).

Would you like a detailed explanation of any step, or do you have any questions?

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Here are five related questions you might consider:
1. How do you solve a quadratic equation using the quadratic formula?
2. What are the steps to find a common denominator for adding fractions?
3. How do you check if a solution to a quadratic equation is correct?
4. What are the key differences between solving linear and quadratic equations?
5. How do you factorize quadratic expressions?

**Tip:** Always check if the solutions satisfy the original equation, especially in rational equations, where extraneous solutions may arise.

Learn how to solve the rational equation (2/(x-3)) - (3/(x-2)) = (2/3) - (3/2) step-by-step. This problem involves equations with fractions and requires understanding of algebraic manipulation and the quadratic formula.

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