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The given inequality is:

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

### Steps to solve:

1. **Isolate the variable**: First, we can eliminate the fraction by multiplying both sides of the inequality by \(2x + 2\). Before doing so, we should check whether the expression \(2x + 2\) is positive or negative because multiplying by a negative number reverses the inequality.

   Let's first solve for when \(2x + 2 = 0\), i.e.,

   \[
   2x + 2 = 0 \quad \Rightarrow \quad x = -1
   \]

   This means:
   - For \(x > -1\), \(2x + 2 > 0\) (positive), so the inequality sign remains the same.
   - For \(x < -1\), \(2x + 2 < 0\) (negative), so the inequality sign will reverse.

2. **Multiply by \(2x + 2\)**:
   For \(x > -1\):
   \[
   1 > -4(2x + 2)
   \]

   Distribute the \(-4\) on the right-hand side:
   \[
   1 > -8x - 8
   \]

   Now, solve for \(x\):

   \[
   1 + 8 > -8x
   \]

   \[
   9 > -8x \quad \Rightarrow \quad x > -\frac{9}{8}
   \]

3. **Conclusion**:
   For \(x > -1\), the solution is \(x > -\frac{9}{8}\).

   For \(x < -1\), we would need to reverse the inequality sign after multiplying, but further work would show that no solutions exist in that region. Hence, the final solution is:

\[
x > -1
\]

---

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

Here are 5 related questions to expand this topic:
1. What happens to an inequality when you multiply both sides by a negative number?
2. How do you solve inequalities involving absolute values?
3. How can we graph the solution \(x > -1\) on a number line?
4. How do inequalities with rational expressions differ from linear inequalities?
5. What strategies can be used to solve complex inequalities involving multiple fractions?

**Tip:** Always check for when the denominator equals zero in rational inequalities, as this can affect the solution.

Find all values of x which satisfy the following inequality: 1/(2x + 2) > -4

Learn how to solve the inequality 1/(2x + 2) > -4 step by step. This algebra problem includes concepts of rational expressions and inequalities, and highlights how the sign of an inequality changes when multiplying by a negative. Suitable for students in Grades 9-11.

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