Let's solve the inequality step-by-step and then graph the solution set on the number line.
First, consider the inequality [tex]\( -2x + 9 \leq 5x - 12 \)[/tex].
1. Rearrange the inequality to get all [tex]\( x \)[/tex]-terms on one side and the constants on the other side:
[tex]\[
-2x + 9 \leq 5x - 12
\][/tex]
Add [tex]\( 2x \)[/tex] to both sides:
[tex]\[
9 \leq 7x - 12
\][/tex]
Add 12 to both sides:
[tex]\[
21 \leq 7x
\][/tex]
2. Solve for [tex]\( x \)[/tex] by dividing both sides by 7:
[tex]\[
\frac{21}{7} \leq x
\][/tex]
Simplify:
[tex]\[
3 \leq x
\][/tex]
which can also be written as:
[tex]\[
x \geq 3
\][/tex]
The solution to the inequality [tex]\( -2x + 9 \leq 5x - 12 \)[/tex] is [tex]\( x \geq 3 \)[/tex].
Now, let's graph this solution set on a number line:
1. Draw a horizontal line to represent the number line.
2. Mark a point on the number line at [tex]\( x = 3 \)[/tex].
3. Since [tex]\( x \)[/tex] is greater than or equal to 3, we:
- Place a closed or filled-in circle at [tex]\( x = 3 \)[/tex] to indicate that [tex]\( x = 3 \)[/tex] is included in the solution set.
- Shade the number line to the right of [tex]\( x = 3 \)[/tex] to indicate that all numbers greater than or equal to 3 are included in the solution set.
Here is the representation:
```
<-----------------------------|------------------------------>
... 0 1 2 | 3 4 5 ...
(closed circle at 3)
and arrow pointing right from 3
```
This shaded number line represents the solution set [tex]\( x \geq 3 \)[/tex].
