The given compound inequality is:

\[
4x - 7 \leq -27 \quad \text{or} \quad 29 > 4x - 7
\]

Let's solve each part of the inequality separately.

### Part 1: \(4x - 7 \leq -27\)

1. Add 7 to both sides:
   \[
   4x \leq -27 + 7
   \]
   \[
   4x \leq -20
   \]

2. Divide both sides by 4:
   \[
   x \leq -5
   \]

### Part 2: \(29 > 4x - 7\)

1. Add 7 to both sides:
   \[
   29 + 7 > 4x
   \]
   \[
   36 > 4x
   \]

2. Divide both sides by 4:
   \[
   9 > x \quad \text{or} \quad x < 9
   \]

### Final Solution

The solution to the compound inequality is:

\[
x \leq -5 \quad \text{or} \quad x < 9
\]

This solution means \(x\) can be any value less than or equal to \(-5\), or any value less than \(9\). Therefore, the combined solution set is:

\[
(-\infty, -5] \cup (-\infty, 9)
\]

Since \(x < 9\) covers all values below \(9\), the overall solution can be simplified to:

\[
x < 9
\]

### Do you want any more details or have questions?

Here are 5 related questions to further explore this topic:
1. How do we graph the solution \(x < 9\) on a number line?
2. What if the inequality involved "and" instead of "or"?
3. How would the solution change if the inequality was \(4x - 7 \geq -27\)?
4. What happens if both sides of the inequality were multiplied by a negative number?
5. Can we apply similar steps to solve inequalities involving fractions?

**Tip:** Always be careful with the inequality signs when performing operations like addition, subtraction, multiplication, or division, especially when dealing with negative numbers.

Learn how to solve the compound inequality 4x - 7 ≤ -27 or 29 > 4x - 7 step-by-step with detailed explanations. This problem covers solving inequalities, including addition, subtraction, multiplication, and division operations. Suitable for students in Grades 7-9.

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