Let's solve the inequality step by step.
1. Start with the given inequality:
[tex]\[
\frac{1}{3}(9x + 27) > x + 33
\][/tex]
2. Distribute the [tex]\(\frac{1}{3}\)[/tex] on the left side:
[tex]\[
\frac{1}{3} \cdot 9x + \frac{1}{3} \cdot 27 > x + 33
\][/tex]
Which simplifies to:
[tex]\[
3x + 9 > x + 33
\][/tex]
3. Move all terms involving [tex]\(x\)[/tex] to one side and constant terms to the other side of the inequality. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
3x - x + 9 > 33
\][/tex]
Simplifying, we get:
[tex]\[
2x + 9 > 33
\][/tex]
4. Move the constant term [tex]\(9\)[/tex] to the other side:
[tex]\[
2x > 33 - 9
\][/tex]
Which simplifies to:
[tex]\[
2x > 24
\][/tex]
5. Divide both sides by 2:
[tex]\[
x > 12
\][/tex]
The solution to the inequality is [tex]\(x > 12\)[/tex].
To represent this on a graph,
- The graph should have an open circle at [tex]\(x = 12\)[/tex] (indicating that 12 is not included in the solution) and a shaded region to the right of 12.
Therefore, the correct graph representation of the solution [tex]\(x > 12\)[/tex] corresponds to an open circle at 12 and shading to the right, indicating all values greater than 12.
Choose the letter that corresponds to this description from the given options A, B, C, or D. In this case, D is the correct choice.
