Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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.
### 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.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.