Answer :
Sure, let's solve the inequality step-by-step and then graph the solution set on a number line.
### Solving the Inequality
1. Start with the inequality:
[tex]\[ -2x + 9 \leq 5x - 12 \][/tex]
2. Combine like terms by adding \(2x\) to both sides:
[tex]\[ 9 \leq 7x - 12 \][/tex]
3. Add 12 to both sides to isolate the term with \(x\):
[tex]\[ 9 + 12 \leq 7x \][/tex]
[tex]\[ 21 \leq 7x \][/tex]
4. Divide both sides by 7 to solve for \(x\):
[tex]\[ x \geq 3 \][/tex]
So the solution to the inequality is:
[tex]\[ x \geq 3 \][/tex]
### Graphing the Solution Set on the Number Line
To graph the solution \( x \geq 3 \) on the number line:
- Draw a number line.
- Identify the point 3 on the number line.
- Since \(x\) is greater than or equal to 3, you will draw a closed (or filled) circle at 3. This indicates that 3 is included in the solution set.
- Shade or draw an arrow to the right of 3 to show that all values greater than 3 are included in the solution set.
Here is a visual representation of the number line:
```
<---|---|---|---|>
0 1 2 3 4
```
- The closed circle at 3 indicates that the point 3 is included.
- The arrow or continuous line to the right of 3 indicates that every number greater than 3 is a part of the solution set.
This visual representation on the number line effectively shows [tex]\( x \geq 3 \)[/tex].
### Solving the Inequality
1. Start with the inequality:
[tex]\[ -2x + 9 \leq 5x - 12 \][/tex]
2. Combine like terms by adding \(2x\) to both sides:
[tex]\[ 9 \leq 7x - 12 \][/tex]
3. Add 12 to both sides to isolate the term with \(x\):
[tex]\[ 9 + 12 \leq 7x \][/tex]
[tex]\[ 21 \leq 7x \][/tex]
4. Divide both sides by 7 to solve for \(x\):
[tex]\[ x \geq 3 \][/tex]
So the solution to the inequality is:
[tex]\[ x \geq 3 \][/tex]
### Graphing the Solution Set on the Number Line
To graph the solution \( x \geq 3 \) on the number line:
- Draw a number line.
- Identify the point 3 on the number line.
- Since \(x\) is greater than or equal to 3, you will draw a closed (or filled) circle at 3. This indicates that 3 is included in the solution set.
- Shade or draw an arrow to the right of 3 to show that all values greater than 3 are included in the solution set.
Here is a visual representation of the number line:
```
<---|---|---|---|>
0 1 2 3 4
```
- The closed circle at 3 indicates that the point 3 is included.
- The arrow or continuous line to the right of 3 indicates that every number greater than 3 is a part of the solution set.
This visual representation on the number line effectively shows [tex]\( x \geq 3 \)[/tex].
