What Kind Of Compound Inequality Is $1-4x \ \textless \ 21$ And $5x + 2 \ \textgreater \ 22$?A. Intersection B. Union

Feb 28, 2025 by ADMIN 129 views

$What kind of compound inequality is $1-4x \ \textless \ 21$ and $5x + 2 \ \textgreater \ 22$?$

Introduction to Compound Inequalities

Compound inequalities are a combination of two or more inequalities that are combined using logical operators such as "and" or "or". They are used to describe a range of values that satisfy multiple conditions. In this article, we will explore the type of compound inequality represented by the given inequalities $1-4x \ \textless \ 21$ and $5x + 2 \ \textgreater \ 22$.

Understanding the Inequalities

To determine the type of compound inequality, we need to understand the individual inequalities. The first inequality is $1-4x \ \textless \ 21$, which can be rewritten as $-4x \ \textless \ 20$. This inequality can be further simplified by dividing both sides by -4, resulting in $x \ \textgreater \ -5$. The second inequality is $5x + 2 \ \textgreater \ 22$, which can be rewritten as $5x \ \textgreater \ 20$. Dividing both sides by 5, we get $x \ \textgreater \ 4$.

Intersection vs Union

There are two types of compound inequalities: intersection and union. The intersection of two inequalities is the set of values that satisfy both inequalities simultaneously. The union of two inequalities is the set of values that satisfy at least one of the inequalities.

Analyzing the Inequalities

To determine the type of compound inequality, we need to analyze the individual inequalities and their intersection. The first inequality $x \ \textgreater \ -5$ represents all values greater than -5, while the second inequality $x \ \textgreater \ 4$ represents all values greater than 4. The intersection of these two inequalities is the set of values that satisfy both inequalities simultaneously, which is the set of values greater than 4.

Conclusion

Based on the analysis, the compound inequality $1-4x \ \textless \ 21$ and $5x + 2 \ \textgreater \ 22$ is an intersection of two inequalities. The intersection of the two inequalities is the set of values that satisfy both inequalities simultaneously, which is the set of values greater than 4.

Final Answer

The final answer is A. Intersection.

Additional Information

Compound inequalities are used in various mathematical applications, including algebra, calculus, and statistics. Understanding the type of compound inequality is essential in solving mathematical problems and making informed decisions.

Example Problems

Solve the compound inequality $2x - 3 \ \textless \ 12$ and $x + 2 \ \textgreater \ 8$.
Determine the type of compound inequality represented by the inequalities $x - 2 \ \textless \ 10$ and $3x + 1 \ \textgreater \ 14$.

Solutions to Example Problems

To solve the compound inequality $2x - 3 \ \textless \ 12$ and $x + 2 \ \textgreater \ 8$, we need to analyze the individual inequalities. The first inequality can be rewritten as $2x \ \textless \ 15$, which simplifies to $x \ \textless \ 7.5$. The second inequality can be rewritten as $x \ \textgreater \ 6$. The intersection of these two inequalities is the set of values that satisfy both inequalities simultaneously, which is the set of values greater than 6 and less than 7.5.
To determine the type of compound inequality represented by the inequalities $x - 2 \ \textless \ 10$ and $3x + 1 \ \textgreater \ 14$, we need to analyze the individual inequalities. The first inequality can be rewritten as $x \ \textless \ 12$, while the second inequality can be rewritten as $3x \ \textgreater \ 13$, which simplifies to $x \ \textgreater \ 4.33$. The intersection of these two inequalities is the set of values that satisfy both inequalities simultaneously, which is the set of values greater than 4.33 and less than 12.

Conclusion

In conclusion, compound inequalities are used to describe a range of values that satisfy multiple conditions. Understanding the type of compound inequality is essential in solving mathematical problems and making informed decisions. The intersection of two inequalities is the set of values that satisfy both inequalities simultaneously, while the union of two inequalities is the set of values that satisfy at least one of the inequalities.

Introduction

Compound inequalities are a combination of two or more inequalities that are combined using logical operators such as "and" or "or". They are used to describe a range of values that satisfy multiple conditions. In this article, we will answer some frequently asked questions about compound inequalities.

Q1: What is a compound inequality?

A1: A compound inequality is a combination of two or more inequalities that are combined using logical operators such as "and" or "or".

Q2: What are the two types of compound inequalities?

A2: The two types of compound inequalities are intersection and union. The intersection of two inequalities is the set of values that satisfy both inequalities simultaneously, while the union of two inequalities is the set of values that satisfy at least one of the inequalities.

Q3: How do I solve a compound inequality?

A3: To solve a compound inequality, you need to analyze the individual inequalities and their intersection. You can rewrite the inequalities in a simpler form and then find the intersection of the two inequalities.

Q4: What is the difference between an intersection and a union of inequalities?

A4: The intersection of two inequalities is the set of values that satisfy both inequalities simultaneously, while the union of two inequalities is the set of values that satisfy at least one of the inequalities.

Q5: How do I determine the type of compound inequality?

A5: To determine the type of compound inequality, you need to analyze the individual inequalities and their intersection. If the intersection of the two inequalities is the set of values that satisfy both inequalities simultaneously, then it is an intersection of inequalities. If the intersection of the two inequalities is the set of values that satisfy at least one of the inequalities, then it is a union of inequalities.

Q6: Can I have more than two inequalities in a compound inequality?

A6: Yes, you can have more than two inequalities in a compound inequality. However, the analysis of the compound inequality becomes more complex as the number of inequalities increases.

Q7: How do I graph a compound inequality?

A7: To graph a compound inequality, you need to graph the individual inequalities and then find the intersection of the two inequalities. The intersection of the two inequalities represents the set of values that satisfy both inequalities simultaneously.

Q8: Can I have a compound inequality with both "and" and "or" operators?

A8: Yes, you can have a compound inequality with both "and" and "or" operators. However, the analysis of the compound inequality becomes more complex as the number of operators increases.

Q9: How do I solve a compound inequality with absolute values?

A9: To solve a compound inequality with absolute values, you need to analyze the individual inequalities and their intersection. You can rewrite the inequalities in a simpler form and then find the intersection of the two inequalities.

Q10: Can I have a compound inequality with fractions?

A10: Yes, you can have a compound inequality with fractions. However, the analysis of the compound inequality becomes more complex as the number of fractions increases.

Conclusion

Additional Resources

Final Thoughts

Compound inequalities are a powerful tool in mathematics, and understanding them is essential in solving mathematical problems and making informed decisions. By following the steps outlined in this article, you can solve compound inequalities and make informed decisions in a variety of mathematical applications.