Let's solve the inequality step by step. The problem contains a compound inequality with "or," meaning we need to solve both parts separately.

1st Inequality:

$-24 \leq -3x - 3$

First, isolate the term with $x$: $-24 + 3 \leq -3x$ $-21 \leq -3x$

Now, divide by $-3$ (remember, dividing by a negative number flips the inequality): $7 \geq x \quad \text{or} \quad x \leq 7$

2nd Inequality:

$-39 > -3x - 3$

Isolate $x$ again: $-39 + 3 > -3x$ $-36 > -3x$

Divide by $-3$ and flip the inequality: $12 < x \quad \text{or} \quad x > 12$

Final Solution:

• From the first inequality: $x \leq 7$
• From the second inequality: $x > 12$

Since this is an "or" inequality, the solution is $x \leq 7$ or $x > 12$. In inequality notation, the solution can be written as: $x \in (-\infty, 7] \cup (12, \infty)$

On a number line, we would graph:

• A closed circle at $x = 7$ with a line extending left to negative infinity.
• An open circle at $x = 12$ with a line extending right to positive infinity.

Would you like further details on the graphing process or explanations? Here are some follow-up questions to consider:

1. How does dividing or multiplying by a negative number affect inequalities?
2. What does the term "compound inequality" mean?
3. Can you explain when to use "and" vs. "or" in inequalities?
4. How do we interpret closed and open circles on a number line?
5. What other methods can we use to solve compound inequalities?

Tip: Always remember to flip the inequality when dividing or multiplying by a negative number.