To solve the compound inequality:

$7k + 1 \leq 8 \quad \text{or} \quad -7 < k - 10$

We will solve each part separately.

1. Solve $7k + 1 \leq 8$:

$7k + 1 \leq 8$

Subtract 1 from both sides: $7k \leq 7$

Divide both sides by 7: $k \leq 1$

2. Solve $-7 < k - 10$:

$-7 < k - 10$

Add 10 to both sides: $3 < k$

This can also be written as: $k > 3$

3. Combine the results:

The solution is $k \leq 1$ or $k > 3$.

In interval notation, the solution is: $(-\infty, 1] \cup (3, \infty)$

Would you like more details or have any questions?

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Related Questions:

1. How do you solve compound inequalities involving both "and" & "or"?
2. What is interval notation and how do you use it?
3. How do you graph the solution to a compound inequality on a number line?
4. What is the difference between strict and non-strict inequalities?
5. How do you solve inequalities involving absolute values?

Tip: When dealing with inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number.