Let's solve the inequality step by step and then graph the solutions on a number line.
### Step-by-Step Solution:
1. Start with the inequality:
[tex]\[
6 - 2(x - 3) > 3x - 3
\][/tex]
2. Distribute the [tex]\(-2\)[/tex] within the parenthesis:
[tex]\[
6 - 2x + 6 > 3x - 3
\][/tex]
3. Combine the constant terms on the left side:
[tex]\[
12 - 2x > 3x - 3
\][/tex]
4. Add [tex]\(2x\)[/tex] to both sides to start isolating the variable [tex]\(x\)[/tex]:
[tex]\[
12 > 5x - 3
\][/tex]
5. Add [tex]\(3\)[/tex] to both sides to further isolate the variable [tex]\(x\)[/tex]:
[tex]\[
15 > 5x
\][/tex]
6. Divide both sides by [tex]\(5\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
3 > x
\][/tex]
7. Rewrite the inequality for clarity:
[tex]\[
x < 3
\][/tex]
### Solution Interpretation:
The inequality [tex]\( x < 3 \)[/tex] means that [tex]\( x \)[/tex] can be any number less than [tex]\( 3 \)[/tex], but it does not include [tex]\( 3 \)[/tex] itself.
### Graphing on the Number Line:
1. Draw a number line.
2. Locate the point [tex]\( 3 \)[/tex] on the number line.
3. Draw an open circle at [tex]\( 3 \)[/tex] to indicate that [tex]\( x = 3 \)[/tex] is not included in the solution.
4. Shade the portion of the number line to the left of [tex]\( 3 \)[/tex] to represent all numbers less than [tex]\( 3 \)[/tex].
Here is a sketch of the number line with the solution:
```
-∞ <----------------------------------------------------------(3)
```
The open circle at [tex]\( 3 \)[/tex] indicates that [tex]\( 3 \)[/tex] itself is not included in the set of solutions, and the shading to the left represents all numbers less than [tex]\( 3 \)[/tex].
