To solve the inequality [tex]\( \frac{1}{2}(20x + 6) \geq x + 30 \)[/tex], follow these steps:
1. Distribute the [tex]\(\frac{1}{2}\)[/tex] on the left side:
[tex]\[
\frac{1}{2}(20x + 6) = \frac{1}{2} \cdot 20x + \frac{1}{2} \cdot 6 = 10x + 3
\][/tex]
So the inequality becomes:
[tex]\[
10x + 3 \geq x + 30
\][/tex]
2. Isolate [tex]\(x\)[/tex] on one side of the inequality:
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
10x + 3 - x \geq x + 30 - x
\][/tex]
[tex]\[
9x + 3 \geq 30
\][/tex]
3. Subtract 3 from both sides:
[tex]\[
9x + 3 - 3 \geq 30 - 3
\][/tex]
[tex]\[
9x \geq 27
\][/tex]
4. Divide both sides by 9:
[tex]\[
x \geq \frac{27}{9}
\][/tex]
[tex]\[
x \geq 3
\][/tex]
The solution to the inequality [tex]\( \frac{1}{2}(20x + 6) \geq x + 30 \)[/tex] is [tex]\( x \geq 3 \)[/tex].
Now, to visualize this solution on a number line:
- Draw a number line with a point at [tex]\( x = 3 \)[/tex].
- Since [tex]\( x \geq 3 \)[/tex], you will shade the number line to the right of [tex]\( x = 3 \)[/tex].
- Include a closed circle at [tex]\( 3 \)[/tex] to indicate that [tex]\( x = 3 \)[/tex] is part of the solution set.
This number line representation shows all values of [tex]\( x \)[/tex] that satisfy the inequality [tex]\( x \geq 3 \)[/tex].
