Let's solve the inequality step by step:
1. Rewrite the Inequality:
[tex]\[
-2x + 9 \leq 5x - 12
\][/tex]
2. Move all [tex]\(x\)[/tex] terms to one side and constant terms to the other side:
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[
-2x - 5x + 9 \leq -12
\][/tex]
Simplify the terms:
[tex]\[
-7x + 9 \leq -12
\][/tex]
3. Move the constant term to the other side:
Subtract 9 from both sides:
[tex]\[
-7x \leq -12 - 9
\][/tex]
Simplify the terms:
[tex]\[
-7x \leq -21
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by -7 (remember to reverse the inequality sign when dividing by a negative number):
[tex]\[
x \geq 3
\][/tex]
5. Graph the solution set:
On a number line:
- Draw a line.
- Locate the point [tex]\(3\)[/tex] on the line.
- Since the inequality is [tex]\(\geq\)[/tex], you will use a solid dot at [tex]\(3\)[/tex] to include [tex]\(3\)[/tex] in the solution set.
- Shade the line to the right of [tex]\(3\)[/tex] to represent all numbers greater than or equal to [tex]\(3\)[/tex].
Here's a visual representation of the number line:
```
<-------------------|------------------->
-∞ 0 +∞
|
|-----•=====================>
1 2 3 4 5 ...
```
The solid dot at 3 indicates that [tex]\(3\)[/tex] is included in the solution set, and the arrow to the right shows all numbers greater than or equal to [tex]\(3\)[/tex].
