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The problem requires solving the logarithmic equation:

\[
\log_{1/2}(x^2 + x) - \log_{1/2}(x^2 - x) = -2
\]

### Step-by-Step Solution

1. **Combine the Logarithms**: Use the property of logarithms:
   \[
   \log_a \frac{M}{N} = \log_a M - \log_a N
   \]
   to combine the two logs on the left side.
   \[
   \log_{1/2} \left( \frac{x^2 + x}{x^2 - x} \right) = -2
   \]

2. **Rewrite the Equation in Exponential Form**: Since \(\log_{1/2} A = -2\) is equivalent to \(A = (1/2)^{-2}\), we can rewrite the equation as:
   \[
   \frac{x^2 + x}{x^2 - x} = (1/2)^{-2}
   \]

3. **Simplify the Right Side**: \((1/2)^{-2}\) is the same as \(4\), so the equation becomes:
   \[
   \frac{x^2 + x}{x^2 - x} = 4
   \]

4. **Cross Multiply**:
   \[
   x^2 + x = 4(x^2 - x)
   \]
   Expanding the right side:
   \[
   x^2 + x = 4x^2 - 4x
   \]

5. **Move All Terms to One Side**:
   \[
   0 = 3x^2 - 5x
   \]
   This simplifies to:
   \[
   3x^2 - 5x = 0
   \]

6. **Factor Out \(x\)**:
   \[
   x(3x - 5) = 0
   \]
   So, \(x = 0\) or \(3x - 5 = 0\).

7. **Solve for \(x\)**:
   - If \(x = 0\), then one solution is \(x = 0\).
   - For \(3x - 5 = 0\):
     \[
     3x = 5 \Rightarrow x = \frac{5}{3}
     \]

### Verify Solutions
We must ensure that the solutions do not make any argument of the original logarithms negative or zero:
- For \(x = 0\):
  - \(x^2 + x = 0\), which is invalid in a logarithmic expression.
- For \(x = \frac{5}{3}\):
  - Both \(x^2 + x\) and \(x^2 - x\) are positive, so this is a valid solution.

### Conclusion
The only valid solution is:

\[
x = \frac{5}{3}
\]

**Answer**: The solution set is \(\left\{ \frac{5}{3} \right\}\).

Would you like more details, or do you have any questions about any specific steps?

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### Further Practice Questions
1. Solve \(\log_{2}(x^2 + 3x) - \log_{2}(x^2 - x) = 1\).
2. Simplify and solve: \(\log_{3}(3x^2 + x) - \log_{3}(x + 1) = 2\).
3. Solve for \(x\): \(\log_{4}(x^2 + 4) - \log_{4}(x) = 1\).
4. Prove that if \(\log_{a}(f(x)) - \log_{a}(g(x)) = c\), then \(f(x) = g(x) \cdot a^c\).
5. Solve the equation \(\log_{10}(x^2 - 3x + 2) = 0\).

### Tip
Always check the arguments of logarithmic expressions to avoid extraneous solutions that result in undefined values (like negatives or zero).

Solve the logarithmic equation: log_{1/2}(x^2 + x) - log_{1/2}(x^2 - x) = -2

Learn how to solve the logarithmic equation log_{1/2}(x^2 + x) - log_{1/2}(x^2 - x) = -2. This problem covers key concepts such as properties of logarithms and solving exponential equations. Suitable for students in Grades 11-12.

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