Solve for xx and graph the solution on the number line below.
3, x, minus, 6, is greater than, minus, 21, and, 3, is greater than, 3, x, minus, 6
3x−6>−21and
3>3x−6

Let's solve the inequalities step by step.

### 1. Solve the first inequality: 
\[ 3x - 6 > -21 \]

Add 6 to both sides to isolate the term with \( x \):
\[
3x - 6 + 6 > -21 + 6
\]
\[
3x > -15
\]

Now, divide both sides by 3:
\[
x > -5
\]

### 2. Solve the second inequality: 
\[ 3 > 3x - 6 \]

Add 6 to both sides to isolate the term with \( x \):
\[
3 + 6 > 3x - 6 + 6
\]
\[
9 > 3x
\]

Now, divide both sides by 3:
\[
3 > x
\]
This can also be written as:
\[ x < 3 \]

### 3. Combine the results:
From the two inequalities, we have:
\[
-5 < x < 3
\]
This is the solution set.

### 4. Graph the solution on the number line:
- Draw a number line.
- Mark the points \( -5 \) and \( 3 \) on the number line.
- Draw an open circle at \( -5 \) (since \( x > -5 \)) and an open circle at \( 3 \) (since \( x < 3 \)).
- Shade the region between \( -5 \) and \( 3 \) to represent all the values \( x \) can take.

Would you like me to generate this number line graph for you?

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Here are some related questions you might consider:
1. What is the difference between strict inequalities and non-strict inequalities?
2. How would the solution change if the inequalities were non-strict (i.e., \( \geq \) or \( \leq \))?
3. Can you solve a system of inequalities with more than two conditions?
4. How would you express the solution set of \( -5 < x < 3 \) using interval notation?
5. What does it mean when there is no overlap between the solutions of two inequalities?

**Tip:** When solving compound inequalities, always pay attention to the direction of the inequality signs to combine the solutions correctly.

Learn how to solve the compound inequalities 3x - 6 > -21 and 3 > 3x - 6 step-by-step with clear explanations and graphing the solutions on a number line. Suitable for Grades 7-9 mathematics curriculum.

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