To solve the given compound inequality [tex]\( 4p + 1 < -11 \)[/tex] or [tex]\( 6p + 3 > 39 \)[/tex], we need to solve each inequality separately and then combine the results.
### Step-by-Step Solution
#### Solving the First Inequality: [tex]\( 4p + 1 < -11 \)[/tex]
1. Subtract 1 from both sides of the inequality:
[tex]\[
4p + 1 - 1 < -11 - 1
\][/tex]
[tex]\[
4p < -12
\][/tex]
2. Divide both sides by 4:
[tex]\[
\frac{4p}{4} < \frac{-12}{4}
\][/tex]
[tex]\[
p < -3
\][/tex]
Thus, the solution to the first inequality is:
[tex]\[
p < -3
\][/tex]
#### Solving the Second Inequality: [tex]\( 6p + 3 > 39 \)[/tex]
1. Subtract 3 from both sides of the inequality:
[tex]\[
6p + 3 - 3 > 39 - 3
\][/tex]
[tex]\[
6p > 36
\][/tex]
2. Divide both sides by 6:
[tex]\[
\frac{6p}{6} > \frac{36}{6}
\][/tex]
[tex]\[
p > 6
\][/tex]
Thus, the solution to the second inequality is:
[tex]\[
p > 6
\][/tex]
### Combining the Solutions
The compound inequality involves "or" ([tex]\( \cup \)[/tex]), so the solution is the union of the two individual solutions:
[tex]\[
p < -3 \quad \text{or} \quad p > 6
\][/tex]
### Graph of the Solution
To graph this solution, we represent the intervals:
- For [tex]\( p < -3 \)[/tex]:
An open circle at [tex]\( p = -3 \)[/tex] indicating that [tex]\(-3\)[/tex] is not included, and shading to the left.
- For [tex]\( p > 6 \)[/tex]:
An open circle at [tex]\( p = 6 \)[/tex] indicating that [tex]\(6\)[/tex] is not included, and shading to the right.
Visually on a number line, it looks like this:
[tex]\[
\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
& & & & & & -3 & & & 6 & & & & & \\
\hline
& & & & & \circ &\leftarrow&\leftarrow&\leftarrow& &\rightarrow&\rightarrow&\circ& & & & \\
\end{array}
\][/tex]
The graph shows:
- Numbers to the left of [tex]\(-3\)[/tex] (shaded to the left with an open circle at [tex]\(-3\)[/tex]).
- Numbers to the right of [tex]\(6\)[/tex] (shaded to the right with an open circle at [tex]\(6\)[/tex]).
This is the correct graphical representation of the given compound inequality.
