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.