https://file.aiquickdraw.com/mathbot/file/1724150776801luqiau78.jpg

The equation shown in the image is:

\[
\frac{4}{x - 2} - \frac{3}{x + 1} = 1
\]

### To solve this rational equation:

1. **Isolate the Fractions:**
   \[
   \frac{4}{x - 2} - \frac{3}{x + 1} = 1
   \]

2. **Find a common denominator** for the fractions to combine them. The common denominator for \((x - 2)\) and \((x + 1)\) is \((x - 2)(x + 1)\).

   Multiply each term by \((x - 2)(x + 1)\) to eliminate the denominators:
   \[
   4(x + 1) - 3(x - 2) = (x - 2)(x + 1)
   \]

3. **Simplify the Equation:**
   Expand and simplify both sides:
   \[
   4x + 4 - 3x + 6 = x^2 - x - 2
   \]
   Combine like terms:
   \[
   x + 10 = x^2 - x - 2
   \]

4. **Rearrange the equation to form a quadratic equation:**
   Move all terms to one side of the equation:
   \[
   x^2 - 2x - 12 = 0
   \]

5. **Factor the Quadratic Equation:**
   Find factors of \(-12\) that add up to \(-2\). These factors are \(-6\) and \(2\).
   \[
   (x - 6)(x + 2) = 0
   \]

6. **Solve for \(x\):**
   Set each factor equal to zero:
   \[
   x - 6 = 0 \quad \text{or} \quad x + 2 = 0
   \]
   So,
   \[
   x = 6 \quad \text{or} \quad x = -2
   \]

7. **Check for extraneous solutions:**
   Substitute \(x = 6\) and \(x = -2\) back into the original equation to ensure they do not result in a zero denominator:
   - For \(x = 6\): The equation is valid.
   - For \(x = -2\): This would make the denominator of the first term \(x - 2\) equal to zero, which is undefined.

   Therefore, \(x = -2\) is an extraneous solution and should be discarded.

### Final Answer:
The solution is \(x = 6\).

Would you like more details on any step or have any questions? 

Here are some related questions:

1. What is a rational equation, and how do you solve it in general?
2. Why do we need to check for extraneous solutions when solving rational equations?
3. How would the solution change if the right-hand side of the equation was different?
4. What is the significance of finding a common denominator when solving rational equations?
5. How do you factor quadratic equations efficiently?

**Tip:** Always check for extraneous solutions when solving rational equations, as not all solutions found will necessarily satisfy the original equation due to the possibility of division by zero.

Learn how to solve the rational equation 4/(x - 2) - 3/(x + 1) = 1 with detailed step-by-step instructions. This problem covers key algebraic concepts including rational equations, common denominators, and quadratic equations. Suitable for high school students.

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