The given compound inequality is:

\[
-x + 3 \leq -3 \quad \text{and} \quad -x + 3 > -8
\]

Let's solve each part of the inequality separately.

### 1st inequality: \(-x + 3 \leq -3\)

Subtract 3 from both sides:
\[
-x \leq -6
\]

Now, multiply both sides by \(-1\), remembering to reverse the inequality sign:
\[
x \geq 6
\]

### 2nd inequality: \(-x + 3 > -8\)

Subtract 3 from both sides:
\[
-x > -11
\]

Now, multiply both sides by \(-1\), again reversing the inequality sign:
\[
x < 11
\]

### Final solution:

Now, combine the results from both inequalities:
\[
6 \leq x < 11
\]

Thus, the solution is:
\[
x \in [6, 11)
\]

Would you like more details or have any questions about the steps?

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Here are 5 related questions:

1. How do you solve compound inequalities involving "or" instead of "and"?
2. What happens when multiplying or dividing inequalities by a negative number?
3. How can you graph the solution to this inequality on a number line?
4. What if the compound inequality included variables on both sides?
5. How do you interpret inequalities with absolute values?

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

This guide explains how to solve the compound inequality −x + 3 ≤ −3 and −x + 3 > −8. It covers key algebraic concepts like inequality operations and the rule for reversing inequalities when multiplying by negative numbers. Perfect for Grades 7-9 students.

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