Choose the correct graph.

A.

A number line is labeled from negative 5 to 5 in increments of 1. The entire number line is shaded.-5-4-3-2-1012345

B.

A number line is labeled from 0 to 10 in increments of 1. The region to the left of a right parenthesis at 3 is shaded.012345678910

C.

A number line is labeled from 0 to 10 in increments of 1. The region to the right of a left parenthesis at 2 is shaded.012345678910

D.

A number line is labeled from negative 5 to 5 in increments of 1.-5-4-3-2-1012345

To determine which graph correctly represents the solution to the inequality problem:

**Solution Recap**: We found that the inequality \(6\left(\frac{1}{3}p + 5\right) < 6\left(\frac{1}{3}p - 4\right)\) simplifies to a contradiction, \(30 < -24\), which is always false. Thus, the solution set is empty (\(\emptyset\)).

This means that there are no values of \(p\) that satisfy the inequality, and on a number line, this would be represented by a lack of any shaded region.

**Choosing the Correct Graph:**

Let's examine each option:

- **A**: The entire number line is shaded. This represents all real numbers, which is not correct because there are no solutions.
- **B**: The region to the left of a right parenthesis at 3 is shaded. This suggests a range of numbers less than 3, which is not correct for an empty set.
- **C**: The region to the right of a left parenthesis at 2 is shaded. This suggests a range of numbers greater than 2, which is also not correct for an empty set.
- **D**: The number line is not shaded at all. This represents no solutions, which is correct.

**Conclusion**: The correct graph is **D**, as it represents an empty solution set with no shaded region.

Would you like more details or have any questions about this?

Here are five related questions you might consider:

1. How would the graph look if the inequality had an infinite set of solutions?
2. What are some examples of inequalities that have no solutions?
3. How can we graph inequalities with open and closed endpoints?
4. How would we represent a solution set that includes all real numbers on a number line?
5. How can we verify our graphical solutions for inequalities?

**Tip**: When dealing with inequalities, always check if the solution set is empty by simplifying and verifying the results for any contradictions.

Learn how to choose the correct graph representation for an inequality problem. This problem involves understanding inequalities and their graphical representation on a number line. Explore why certain graphs accurately represent empty solution sets. Suitable for students in Grades 7-9.

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