Let's take the inequality [tex]\(4x > -2x + 24\)[/tex] and solve it step-by-step.
1. Move [tex]\(-2x\)[/tex] to the left side:
To eliminate the variable [tex]\(x\)[/tex] from the right-hand side, we add [tex]\(2x\)[/tex] to both sides of the inequality:
[tex]\[
4x + 2x > 24
\][/tex]
2. Combine like terms:
Simplify the left-hand side by combining the [tex]\(x\)[/tex]-terms:
[tex]\[
6x > 24
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], divide both sides of the inequality by 6:
[tex]\[
x > \frac{24}{6}
\][/tex]
Simplify the fraction:
[tex]\[
x > 4
\][/tex]
The solution to the inequality in inequality notation is:
[tex]\[
x > 4
\][/tex]
4. Interval notation:
In interval notation, the solution set of [tex]\(x > 4\)[/tex] is represented as:
[tex]\[
(4, \infty)
\][/tex]
5. Graphing the solution set:
To graph the solution on a number line:
- Draw a number line with a point at [tex]\(4\)[/tex].
- Since the inequality is [tex]\(x > 4\)[/tex] (strictly greater than), use an open circle at [tex]\(4\)[/tex] to indicate that [tex]\(4\)[/tex] is not included in the solution set.
- Shade the portion of the number line to the right of [tex]\(4\)[/tex] to indicate that all numbers greater than [tex]\(4\)[/tex] are in the solution set.
Here’s what the graph looks like:
```
<----|----|----|----|----|----|----|----|---->
2 3 4 5 6 7 8 9
o----------->
```
The open circle at [tex]\(4\)[/tex] indicates that [tex]\(4\)[/tex] is not part of the solution set, and the shading to the right of [tex]\(4\)[/tex] extends towards infinity, illustrating that all numbers greater than [tex]\(4\)[/tex] satisfy the inequality.
