Let's solve the inequalities step by step.

### 1. Solve \( -3x - 10 > 11 \):
\[
-3x - 10 > 11
\]
First, add 10 to both sides:
\[
-3x > 21
\]
Now, divide both sides by \(-3\). **Remember**, when dividing by a negative number, you must flip the inequality sign:
\[
x < -7
\]
So, the solution to the first inequality is:
\[
x < -7
\]

### 2. Solve \( -34 \leq -3x - 10 \):
\[
-34 \leq -3x - 10
\]
First, add 10 to both sides:
\[
-24 \leq -3x
\]
Now, divide both sides by \(-3\), and **flip the inequality sign**:
\[
8 \geq x \quad \text{or} \quad x \leq 8
\]
So, the solution to the second inequality is:
\[
x \leq 8
\]

### 3. Final Solution:
The inequality is an "or" condition. Therefore, the solution is:
\[
x < -7 \quad \text{or} \quad x \leq 8
\]
Since \(x < -7\) and \(x \leq 8\) both include values, the combined solution is:
\[
x \in (-\infty, -7) \cup (-\infty, 8]
\]

Would you like further explanation on this? 

Here are 5 related questions to consider:
1. What happens if we have "and" instead of "or" in compound inequalities?
2. How do we solve inequalities with absolute values?
3. How do graphing solutions of inequalities look on a number line?
4. What if one of the inequalities was a strict equality, how would it change the solution?
5. What methods can we use to check the correctness of inequality solutions?

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

Explore how to solve the compound inequality −3x−10>11 or −34≤−3x−10 with step-by-step solutions. This problem covers algebraic concepts such as linear inequalities, manipulating inequalities, and understanding the 'or' condition. Suitable for students in Grades 8-10.

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