To solve the compound inequality [tex]\( 4p + 1 > -7 \)[/tex] and [tex]\( 6p + 3 < 33 \)[/tex] and determine its correct graph, we need to approach each inequality step-by-step and then combine the results.
### Step-by-Step Solution
#### Solve the first inequality: [tex]\( 4p + 1 > -7 \)[/tex]
1. Isolate the term with [tex]\( p \)[/tex] by subtracting 1 from both sides:
[tex]\[
4p + 1 - 1 > -7 - 1
\][/tex]
Simplifies to:
[tex]\[
4p > -8
\][/tex]
2. Solve for [tex]\( p \)[/tex] by dividing both sides by 4:
[tex]\[
p > \frac{-8}{4}
\][/tex]
Simplifies to:
[tex]\[
p > -2
\][/tex]
#### Solve the second inequality: [tex]\( 6p + 3 < 33 \)[/tex]
1. Isolate the term with [tex]\( p \)[/tex] by subtracting 3 from both sides:
[tex]\[
6p + 3 - 3 < 33 - 3
\][/tex]
Simplifies to:
[tex]\[
6p < 30
\][/tex]
2. Solve for [tex]\( p \)[/tex] by dividing both sides by 6:
[tex]\[
p < \frac{30}{6}
\][/tex]
Simplifies to:
[tex]\[
p < 5
\][/tex]
#### Combine the inequalities
* Combining the results from both inequalities, we obtain:
[tex]\[
-2 < p < 5
\][/tex]
### Graphing the Compound Inequality
To correctly graph the compound inequality [tex]\( -2 < p < 5 \)[/tex]:
1. Draw a number line.
2. Mark the points [tex]\( -2 \)[/tex] and [tex]\( 5 \)[/tex] on the number line.
3. Since both inequalities are strict (i.e., they do not include equal to), use open circles at [tex]\( -2 \)[/tex] and [tex]\( 5 \)[/tex].
4. Shade the region between these two points to indicate that [tex]\( p \)[/tex] lies between [tex]\( -2 \)[/tex] and [tex]\( 5 \)[/tex].
Hence, the correct graph of the compound inequality [tex]\( 4p + 1 > -7 \)[/tex] and [tex]\( 6p + 3 < 33 \)[/tex] is a number line with open circles at [tex]\( -2 \)[/tex] and [tex]\( 5 \)[/tex], and shading between these points.
