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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.

Solve for x and graph the solution on the number line below: -12 < 3x - 6 and -3 ≥ 3x - 6.

Learn how to solve and graph the linear inequalities -12 < 3x - 6 and -3 ≥ 3x - 6. This problem involves key algebraic concepts and step-by-step solutions, including graphing on a number line and understanding open/closed intervals. Suitable for students in Grades 7-9.

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