Let's solve the given inequalities step by step:

### Inequality 1: 
\[
-13 \leq -x - 7
\]

1. Add 7 to both sides to isolate the term with \(x\):
\[
-13 + 7 \leq -x
\]
\[
-6 \leq -x
\]

2. Multiply both sides by \(-1\), remembering to reverse the inequality:
\[
6 \geq x \quad \text{or} \quad x \leq 6
\]

### Inequality 2:
\[
-16 \geq -x - 7
\]

1. Add 7 to both sides:
\[
-16 + 7 \geq -x
\]
\[
-9 \geq -x
\]

2. Multiply both sides by \(-1\), reversing the inequality:
\[
9 \leq x \quad \text{or} \quad x \geq 9
\]

### Combined Solution:
The solution is the union of the two inequalities:
\[
x \leq 6 \quad \text{or} \quad x \geq 9
\]

Thus, the solution is:
\[
x \in (-\infty, 6] \cup [9, \infty)
\]

Would you like more details or have any further questions?

Here are five related questions:
1. How do you solve compound inequalities?
2. How does multiplying or dividing by a negative number affect an inequality?
3. Can you graph the solution on a number line?
4. What is the difference between "and" and "or" in inequalities?
5. How would this change if the inequalities used strict signs (< or >)?

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

−13 ≤ −x − 7 or −16 ≥ −x − 7

Learn how to solve compound inequalities such as −13 ≤ −x − 7 or −16 ≥ −x − 7. This step-by-step guide demonstrates key algebraic concepts, including reversing inequalities when multiplying by a negative number. The solution involves finding the union of two intervals, suitable for Grades 8-10.

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