To solve the compound inequality \( 7k + 1 \leq 8 \) or \( -7 < -10 \), we need to analyze each part of the inequality separately.

### 1. Solving \( 7k + 1 \leq 8 \):
\[
7k + 1 \leq 8
\]
Subtract 1 from both sides:
\[
7k \leq 7
\]
Divide both sides by 7:
\[
k \leq 1
\]

### 2. Analyzing \( -7 < -10 \):
This inequality states \( -7 \) is less than \( -10 \). However, \( -7 \) is greater than \( -10 \), so this inequality is false.

### Conclusion:
Since the second part \( -7 < -10 \) is false, the solution relies solely on the first part of the inequality. The final solution is:
\[
k \leq 1
\]

Would you like further details or have any questions?

Here are some related questions:
1. What is the difference between "and" and "or" in compound inequalities?
2. How do you solve compound inequalities involving "and"?
3. What does it mean when one part of an "or" inequality is false?
4. Can compound inequalities have no solution?
5. How do you graph the solution of an inequality on a number line?

**Tip:** Always isolate the variable when solving inequalities to simplify the process.

Learn how to solve compound inequalities involving 7k + 1 <= 8 or -7 < -10 with a detailed step-by-step solution. Understand the concepts of compound inequalities and how to analyze each part separately. Suitable for students in Grades 7-9.

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