Let's solve the inequality step by step:

### First inequality: \( 5x - 29 > -34 \)
1. Add 29 to both sides:
   \[
   5x > -34 + 29
   \]
   \[
   5x > -5
   \]

2. Divide both sides by 5:
   \[
   x > \frac{-5}{5}
   \]
   \[
   x > -1
   \]

### Second inequality: \( 2x + 31 < 29 \)
1. Subtract 31 from both sides:
   \[
   2x < 29 - 31
   \]
   \[
   2x < -2
   \]

2. Divide both sides by 2:
   \[
   x < \frac{-2}{2}
   \]
   \[
   x < -1
   \]

### Final solution:
The solution to the inequality is:
\[
x > -1 \quad \text{or} \quad x < -1
\]

This means \( x \) can be any real number except \( x = -1 \), so the solution is:
\[
x \neq -1
\]

Would you like more details or have any other questions? 

Here are five related questions to expand your understanding:
1. How do we handle inequalities involving multiplication or division by negative numbers?
2. What is the difference between "and" vs. "or" when solving compound inequalities?
3. Can you solve the system of inequalities where the conjunction is "and" instead of "or"?
4. How would the solution change if we squared both sides of the inequalities?
5. What are some real-life applications of solving compound inequalities?

**Tip:** When solving inequalities, remember to reverse the inequality sign if you multiply or divide by a negative number.

Explore how to solve the compound inequalities 5x - 29 > -34 and 2x + 31 < 29 step-by-step. Learn key algebraic methods, inequality properties, and understand how to find solutions excluding x = -1. Suitable for students in Grades 8-10.

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