Let's solve the equation:

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

### Step 1: Eliminate the fractions by cross-multiplying:
\[
(x-3) \cdot (-1) = (x+2) \cdot 1
\]

This simplifies to:
\[
-(x-3) = (x+2)
\]

### Step 2: Simplify the equation:
\[
-x + 3 = x + 2
\]

### Step 3: Combine like terms:
Add \( x \) to both sides:
\[
3 = 2x + 2
\]

Now subtract 2 from both sides:
\[
1 = 2x
\]

### Step 4: Solve for \( x \):
\[
x = \frac{1}{2}
\]

### Step 5: Check for restrictions:
The values \( x = 3 \) and \( x = -2 \) would make the denominators zero, but since \( x = \frac{1}{2} \), no restrictions are violated.

Thus, the solution is:
\[
x = \frac{1}{2}
\]

Would you like more details on any step?

Here are 5 related questions for further exploration:
1. What happens if you have different constants in the fractions instead of -1?
2. How can you solve a similar equation if it includes quadratic terms?
3. What are the restrictions on the domain in rational equations?
4. How can you use substitution to solve similar rational equations?
5. How do you solve rational equations with multiple terms on each side?

**Tip:** Always check if the solution violates any restrictions imposed by the denominator!

Explore how to solve the rational equation 1/(x-3) = -1/(x+2) through detailed step-by-step cross-multiplication. This problem delves into algebraic concepts like rational equations and domain restrictions, providing an educational guide for students in Grades 8-10.

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