To solve the compound inequality \( 4x - 1 > -9 \) or \( 7x - 2 \leq 19 \), let's address each part separately.

### First Inequality: \( 4x - 1 > -9 \)
1. Add 1 to both sides:
   \[
   4x > -8
   \]
2. Divide both sides by 4:
   \[
   x > -2
   \]

### Second Inequality: \( 7x - 2 \leq 19 \)
1. Add 2 to both sides:
   \[
   7x \leq 21
   \]
2. Divide both sides by 7:
   \[
   x \leq 3
   \]

### Combined Solution:
Since the original problem is connected by "or", the solution is the union of both individual solutions:

\[
x > -2 \quad \text{or} \quad x \leq 3
\]

This means any \( x \) greater than \(-2\) or any \( x \) less than or equal to \( 3 \) is a solution.

### Final Answer:
\[
x \in (-2, \infty) \quad \text{or} \quad x \in (-\infty, 3]
\]

This can be simplified to \( x \in (-\infty, \infty) \), which implies all real numbers are solutions.

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Would you like more details on any part of this solution, or do you have any other questions?

Here are some related questions:
1. How do you solve compound inequalities with "and" instead of "or"?
2. What are the key differences between "and" and "or" in inequalities?
3. Can compound inequalities have no solution?
4. How do you graph compound inequalities on a number line?
5. What happens if the inequalities are strict (i.e., "<" instead of "≤")?

**Tip:** When solving compound inequalities, always solve each part separately before combining the solutions based on the logical connector ("and" or "or").

Learn how to solve compound inequalities 4x - 1 > -9 or 7x - 2 ≤ 19 with a detailed step-by-step solution. This problem covers key algebraic concepts, including inequalities and compound inequalities. Suitable for students in Grades 7-9.

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