To solve the inequality [tex]\(\frac{1}{2}(20x + 6) \geq x + 30\)[/tex], we can follow these steps:
1. Eliminate the fraction by multiplying both sides by 2:
[tex]\[
2 \cdot \frac{1}{2} (20x + 6) \geq 2 \cdot (x + 30)
\][/tex]
Which simplifies to:
[tex]\[
20x + 6 \geq 2x + 60
\][/tex]
2. Isolate the variable [tex]\(x\)[/tex] on one side:
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[
20x + 6 - 2x \geq 60
\][/tex]
Which simplifies to:
[tex]\[
18x + 6 \geq 60
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Subtract 6 from both sides:
[tex]\[
18x + 6 - 6 \geq 60 - 6
\][/tex]
Which simplifies to:
[tex]\[
18x \geq 54
\][/tex]
Divide both sides by 18:
[tex]\[
\frac{18x}{18} \geq \frac{54}{18}
\][/tex]
Which simplifies to:
[tex]\[
x \geq 3
\][/tex]
So, the solution to the inequality [tex]\(\frac{1}{2}(20x + 6) \geq x + 30\)[/tex] is [tex]\(x \geq 3\)[/tex].
Number Line Representation:
The number line would show all values of [tex]\(x\)[/tex] starting from 3 and extending to the right infinitely (to positive infinity), and would include the number 3 itself because the inequality is [tex]\(\geq\)[/tex] (greater than or equal to).
Visual Representation:
```
----|====>
3
```
Here, the solid dot at 3 indicates that 3 is included in the solution set, and the arrow to the right indicates all numbers greater than 3 are also part of the solution set.
