Let's solve the compound inequality $3 \geq x - 7 > 1$.

First, break it into two separate inequalities and solve each:

1. $3 \geq x - 7$
2. $x - 7 > 1$

Solving $3 \geq x - 7$:

Add 7 to both sides:

$3 + 7 \geq x$

$10 \geq x$

This can be rewritten as:

$x \leq 10$

Solving $x - 7 > 1$:

Add 7 to both sides:

$x - 7 + 7 > 1 + 7$

$x > 8$

Combining the solutions:

$8 < x \leq 10$

Now, let's graph this solution on the number line:

1. The number line will range around 8 and 10.
2. Since $x > 8$, we use an open circle at 8.
3. Since $x \leq 10$, we use a closed circle at 10.
4. Shade the region between 8 and 10.

Here is the number line representation:

$\text{Open circle at 8} \quad (---) \quad \text{Closed circle at 10}$

8 ∘─────────────────● 10

Would you like more details or have any questions about this solution?

Here are 5 questions you might want to ask:

1. How do you graph inequalities on a number line?
2. What is a compound inequality?
3. How do you solve a system of inequalities?
4. Can you explain interval notation for the solution set?
5. How do you verify the solution of an inequality?

Tip: When solving compound inequalities, treat each part separately before combining the solutions for a complete answer.