https://file.aiquickdraw.com/mathbot/file/1726625779313gik6umrr.jpg

The inequality given in the image is:

\[
\frac{1}{2}(p + 7) + \frac{5}{4}(4p - 3) < 9
\]

### Step 1: Simplify each term
We begin by distributing the fractions through the parentheses:

\[
\frac{1}{2}(p + 7) = \frac{1}{2}p + \frac{1}{2} \times 7 = \frac{1}{2}p + \frac{7}{2}
\]
\[
\frac{5}{4}(4p - 3) = \frac{5}{4} \times 4p - \frac{5}{4} \times 3 = 5p - \frac{15}{4}
\]

So the inequality becomes:
\[
\frac{1}{2}p + \frac{7}{2} + 5p - \frac{15}{4} < 9
\]

### Step 2: Combine like terms
We will combine the terms involving \( p \) and the constants. First, convert everything to have the same denominator for the constants.

The common denominator for \( \frac{7}{2} \) and \( \frac{15}{4} \) is 4, so we rewrite:
\[
\frac{7}{2} = \frac{14}{4}
\]
Thus, the inequality is now:
\[
\frac{1}{2}p + 5p + \frac{14}{4} - \frac{15}{4} < 9
\]
Simplify the constants:
\[
\frac{14}{4} - \frac{15}{4} = \frac{-1}{4}
\]
Now the inequality becomes:
\[
\frac{1}{2}p + 5p - \frac{1}{4} < 9
\]

### Step 3: Eliminate fractions
Multiply the entire inequality by 4 to eliminate the fractions:
\[
4 \times \left( \frac{1}{2}p + 5p - \frac{1}{4} \right) < 4 \times 9
\]
This gives:
\[
2p + 20p - 1 < 36
\]
Simplifying further:
\[
22p - 1 < 36
\]

### Step 4: Solve for \( p \)
Add 1 to both sides:
\[
22p < 37
\]
Divide by 22:
\[
p < \frac{37}{22}
\]

### Step 5: Express the solution
The solution set is:
\[
p < \frac{37}{22}
\]

### Interval notation:
\[
(-\infty, \frac{37}{22})
\]

Would you like a graph of this inequality or further details?

### Related questions:
1. How would the solution change if the inequality were \( \leq \) instead of \( < \)?
2. What happens if we solve a similar inequality but with a negative coefficient of \( p \)?
3. Can you verify this inequality by substituting values of \( p \) into the original inequality?
4. What is the graphical representation of the interval \( (-\infty, \frac{37}{22}) \)?
5. How would you handle an inequality involving absolute values?

**Tip**: Always check the inequality by substituting a value from the solution set into the original inequality to verify the correctness.

Solve the inequality: 1/2(p + 7) + 5/4(4p - 3) < 9. Give the solution set in both interval and graph forms.

Learn how to solve the inequality 1/2(p + 7) + 5/4(4p - 3) < 9 with detailed steps. This tutorial covers key algebraic techniques including the distributive property, combining like terms, and solving inequalities. Suitable for students in Grades 9-11.

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