Let's solve the system of inequalities and graph the solution on the number line.

The given system is:

1. $-12 < 3x - 6$
2. $-3 \geq 3x - 6$

Solving the first inequality:

$-12 < 3x - 6$

Add 6 to both sides:

$-12 + 6 < 3x$

$-6 < 3x$

Now, divide both sides by 3:

$-2 < x$

So, the first inequality is:

$x > -2$

Solving the second inequality:

$-3 \geq 3x - 6$

Add 6 to both sides:

$-3 + 6 \geq 3x$

$3 \geq 3x$

Now, divide both sides by 3:

$1 \geq x$

This simplifies to:

$x \leq 1$

Combined Solution:

Now, combine the two inequalities:

$-2 < x \leq 1$

This means $x$ is greater than $-2$ and less than or equal to $1$. In inequality notation, this can be written as:

$(-2, 1]$

Graph on the Number Line:

• An open circle at $-2$ because $x > -2$ (not inclusive).
• A closed circle at $1$ because $x \leq 1$ (inclusive).
• Shade the region between $-2$ and $1$.

Do you have any further questions, or would you like additional details?

Here are some related questions:

1. How do we interpret the open and closed circles on a number line?
2. What is the difference between strict inequality and inequality with equality?
3. How would this change if one inequality was reversed?
4. Can we solve a similar inequality if it included fractions?
5. What if the inequality had more than two conditions?

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