https://file.aiquickdraw.com/mathbot/file/1728347786993b29kr3di.jpg

The given equation in the image is:

\[
\frac{x - 2}{x + 3} = \frac{x - 4}{x + 5}
\]

### Solving algebraically:

1. **Cross-multiply** to eliminate the fractions:
   \[
   (x - 2)(x + 5) = (x - 4)(x + 3)
   \]

2. **Expand both sides**:

   - Left side:
     \[
     (x - 2)(x + 5) = x(x + 5) - 2(x + 5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10
     \]

   - Right side:
     \[
     (x - 4)(x + 3) = x(x + 3) - 4(x + 3) = x^2 + 3x - 4x - 12 = x^2 - x - 12
     \]

3. **Set the expanded forms equal**:
   \[
   x^2 + 3x - 10 = x^2 - x - 12
   \]

4. **Subtract \(x^2\) from both sides** (since they cancel out):
   \[
   3x - 10 = -x - 12
   \]

5. **Move all terms involving \(x\) to one side and constant terms to the other**:
   \[
   3x + x = -12 + 10
   \]
   \[
   4x = -2
   \]

6. **Solve for \(x\)**:
   \[
   x = \frac{-2}{4} = -\frac{1}{2}
   \]

So, the solution is \(x = -\frac{1}{2}\).

### Verifying the solution:

Plug \(x = -\frac{1}{2}\) back into the original equation:

- Left-hand side:
  \[
  \frac{-\frac{1}{2} - 2}{-\frac{1}{2} + 3} = \frac{-\frac{5}{2}}{\frac{5}{2}} = -1
  \]

- Right-hand side:
  \[
  \frac{-\frac{1}{2} - 4}{-\frac{1}{2} + 5} = \frac{-\frac{9}{2}}{\frac{9}{2}} = -1
  \]

Both sides are equal, so the solution is correct.

Would you like any more details or have any questions?

### Related questions:
1. What happens if both sides of the equation have different powers of \(x\)?
2. How do you solve equations with quadratic terms?
3. What is the importance of cross-multiplying in rational equations?
4. How can graphing calculators be used to verify solutions?
5. What are the possible cases where no solution exists in rational equations?

### Tip:
Always check for restrictions (denominators cannot be zero) when solving rational equations.

Solve the equation (x - 2) / (x + 3) = (x - 4) / (x + 5) algebraically and verify the solution.

Learn how to solve the rational equation (x - 2) / (x + 3) = (x - 4) / (x + 5) step-by-step using cross-multiplication. This tutorial guides through the solution and verification process, ideal for high school students in Grades 10-12.

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