Which Graph Represents The Solution Set Of The Compound Inequality Below?$ X + 3 \ \textless \ \frac{1}{2}(4x - 12) \ \textless \ 20 $

Understanding Compound Inequalities

Compound inequalities are a combination of two or more inequalities joined by the words "and" or "or." In this case, we have a compound inequality that involves three expressions: $x + 3$, $\frac{1}{2}(4x - 12)$, and $20$. The inequality is written as $x + 3 < \frac{1}{2}(4x - 12) < 20$. Our goal is to determine which graph represents the solution set of this compound inequality.

Simplifying the Compound Inequality

To simplify the compound inequality, we need to isolate the variable $x$ in each expression. Let's start by simplifying the expression $\frac{1}{2}(4x - 12)$.

$\frac{1}{2}(4x - 12) = 2x - 6$

Now, we can rewrite the compound inequality as:

$x + 3 < 2x - 6 < 20$

Breaking Down the Compound Inequality

To solve the compound inequality, we need to break it down into two separate inequalities:

1. $x + 3 < 2x - 6$
2. $2x - 6 < 20$

Solving the First Inequality

Let's solve the first inequality:

$x + 3 < 2x - 6$

Subtracting $x$ from both sides gives us:

$3 < x - 6$

Adding $6$ to both sides gives us:

$9 < x$

Solving the Second Inequality

Now, let's solve the second inequality:

$2x - 6 < 20$

Adding $6$ to both sides gives us:

$2x < 26$

Dividing both sides by $2$ gives us:

$x < 13$

Combining the Solutions

Now that we have solved both inequalities, we can combine the solutions to find the solution set of the compound inequality.

$9 < x < 13$

Graphing the Solution Set

To graph the solution set, we need to plot the numbers on a number line. The solution set is the interval between $9$ and $13$, not including $9$ and $13$.

Which Graph Represents the Solution Set?

Now that we have the solution set, we can determine which graph represents it. The correct graph is the one that shows the interval between $9$ and $13$, not including $9$ and $13$.

Conclusion

In this article, we have learned how to simplify and solve a compound inequality. We have also learned how to graph the solution set of the compound inequality. By following these steps, we can determine which graph represents the solution set of the compound inequality.

Frequently Asked Questions

• Q: What is a compound inequality? A: A compound inequality is a combination of two or more inequalities joined by the words "and" or "or."
• Q: How do I simplify a compound inequality? A: To simplify a compound inequality, you need to isolate the variable in each expression.
• Q: How do I graph the solution set of a compound inequality? A: To graph the solution set, you need to plot the numbers on a number line.

Final Thoughts

Compound inequalities can be challenging to solve, but with practice and patience, you can master them. Remember to simplify the inequality, break it down into separate inequalities, and graph the solution set. By following these steps, you can determine which graph represents the solution set of the compound inequality.

Additional Resources

• Khan Academy: Compound Inequalities
• Mathway: Compound Inequalities
• Purplemath: Compound Inequalities

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Mathematics for the Nonmathematician" by Morris Kline
  

Understanding Compound Inequalities

Compound inequalities are a combination of two or more inequalities joined by the words "and" or "or." In this article, we will answer some of the most frequently asked questions about compound inequalities.

Q&A

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities joined by the words "and" or "or."

Q: How do I simplify a compound inequality?

A: To simplify a compound inequality, you need to isolate the variable in each expression. This can be done by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

Q: How do I graph the solution set of a compound inequality?

A: To graph the solution set, you need to plot the numbers on a number line. The solution set is the interval between the two values that satisfy the inequality.

Q: What is the difference between a compound inequality and a double inequality?

A: A compound inequality is a combination of two or more inequalities joined by the words "and" or "or." A double inequality is a combination of two inequalities joined by the words "and" only.

Q: How do I solve a compound inequality with fractions?

A: To solve a compound inequality with fractions, you need to eliminate the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.

Q: Can I use a calculator to solve a compound inequality?

A: Yes, you can use a calculator to solve a compound inequality. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct operations.

Q: How do I check my answer to a compound inequality?

A: To check your answer, you need to plug the value back into the original inequality and make sure that it is true.

Q: Can I use a graphing calculator to graph the solution set of a compound inequality?

A: Yes, you can use a graphing calculator to graph the solution set of a compound inequality. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct operations.

Q: How do I write a compound inequality in interval notation?

A: To write a compound inequality in interval notation, you need to use the following notation:

• $(a, b)$ to represent the interval between $a$ and $b$, not including $a$ and $b$
• $[a, b]$ to represent the interval between $a$ and $b$, including $a$ and $b$
• $(a, b]$ to represent the interval between $a$ and $b$, including $b$ but not $a$
• $[a, b)$ to represent the interval between $a$ and $b$, including $a$ but not $b$

Conclusion

In this article, we have answered some of the most frequently asked questions about compound inequalities. We have covered topics such as simplifying compound inequalities, graphing the solution set, and writing compound inequalities in interval notation. By following these steps, you can master compound inequalities and solve them with ease.

Additional Resources

• Khan Academy: Compound Inequalities
• Mathway: Compound Inequalities
• Purplemath: Compound Inequalities

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Mathematics for the Nonmathematician" by Morris Kline