Solve the​ inequality, giving its solution set in both interval and graph form.

6 left parenthesis one third p plus 5 right parenthesis613p+5less than<6 left parenthesis one third p minus 4 right parenthesis613p−4

To solve the inequality, let's first rewrite it for clarity:

\[
6\left(\frac{1}{3}p + 5\right) < 6\left(\frac{1}{3}p - 4\right)
\]

### Step 1: Distribute the 6 on both sides

Distribute the 6 through the parentheses on both sides of the inequality:

\[
6 \cdot \frac{1}{3}p + 6 \cdot 5 < 6 \cdot \frac{1}{3}p - 6 \cdot 4
\]

\[
2p + 30 < 2p - 24
\]

### Step 2: Simplify the inequality

Now, subtract \(2p\) from both sides to isolate the constant terms:

\[
2p + 30 - 2p < 2p - 24 - 2p
\]

\[
30 < -24
\]

### Step 3: Analyze the result

The inequality \(30 < -24\) is false. This suggests that there is no value of \(p\) that will satisfy this inequality.

### Conclusion

Since the inequality is always false, there is no solution set. In interval notation, the solution set is represented as an empty set:

\[
\emptyset
\]

### Graphical Representation

On a number line, the solution set would be shown by having no shaded region or marks, indicating no solutions. It effectively means there is no region where this inequality holds true.

### Summary

- **Interval Notation:** \(\emptyset\) (empty set)
- **Graphical Form:** No region is shaded, indicating no solutions.

Would you like more details on any part of this solution, or do you have any specific questions?

Here are five related questions you might consider:

1. What happens if you change the inequality sign from "<" to ">"?
2. How would the solution change if the constant terms were different?
3. What if the coefficients of \(p\) were different on each side?
4. Can we derive a general approach to solving inequalities involving distribution?
5. How do you represent inequalities on a number line?

**Tip:** Always verify the simplified result of an inequality for any contradiction, as this can indicate no solution exists.

Learn how to solve the inequality 6(1/3p + 5) < 6(1/3p - 4) with a detailed step-by-step solution. This problem involves inequalities and the distributive property. Suitable for students in Grades 9-12.

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