Which Set Of Numbers Is Included In The Solution Set Of The Compound Inequality?A. { {-7, 5, 18, 24, 32}$}$B. { {-9, 7, 15, 22, 26}$}$C. { {16, 17, 22, 23, 24}$}$D. { {18, 19, 20, 21, 22}$}$
Introduction
Compound inequalities are a type of mathematical expression that involves two or more inequalities joined by the words "and" or "or." Solving compound inequalities requires a clear understanding of the individual inequalities and how they interact with each other. In this article, we will explore the concept of compound inequalities and provide a step-by-step guide on how to solve them.
What are Compound Inequalities?
A compound inequality is a mathematical expression that involves two or more inequalities joined by the words "and" or "or." For example:
- 2 < x < 5 and x > 3
- x > 2 or x < 4
Compound inequalities can be solved using various methods, including graphing, algebraic manipulation, and numerical methods.
Types of Compound Inequalities
There are two main types of compound inequalities:
- Conjunction: This type of compound inequality involves two or more inequalities joined by the word "and." For example: 2 < x < 5 and x > 3
- Disjunction: This type of compound inequality involves two or more inequalities joined by the word "or." For example: x > 2 or x < 4
Solving Compound Inequalities
To solve a compound inequality, we need to follow these steps:
- Simplify the individual inequalities: Simplify each individual inequality by combining like terms and eliminating any unnecessary variables.
- Determine the relationship between the inequalities: Determine the relationship between the two inequalities, such as whether they are joined by "and" or "or."
- Graph the inequalities: Graph each individual inequality on a number line or coordinate plane.
- Find the intersection of the inequalities: Find the intersection of the two inequalities, which represents the solution set.
- Write the solution set: Write the solution set in interval notation or as a list of numbers.
Example 1: Solving a Conjunction Compound Inequality
Let's consider the compound inequality: 2 < x < 5 and x > 3
To solve this inequality, we need to follow the steps outlined above:
- Simplify the individual inequalities: The individual inequalities are already simplified.
- Determine the relationship between the inequalities: The inequalities are joined by "and."
- Graph the inequalities: Graph each individual inequality on a number line or coordinate plane.
The graph of the first inequality is a number line with the points 2 and 5 marked. The graph of the second inequality is a number line with the point 3 marked.
- Find the intersection of the inequalities: The intersection of the two inequalities is the region where the two number lines overlap.
The intersection of the two inequalities is the region between 3 and 5, inclusive.
- Write the solution set: The solution set is the region between 3 and 5, inclusive, which can be written as: 3 ≤ x ≤ 5
Example 2: Solving a Disjunction Compound Inequality
Let's consider the compound inequality: x > 2 or x < 4
To solve this inequality, we need to follow the steps outlined above:
- Simplify the individual inequalities: The individual inequalities are already simplified.
- Determine the relationship between the inequalities: The inequalities are joined by "or."
- Graph the inequalities: Graph each individual inequality on a number line or coordinate plane.
The graph of the first inequality is a number line with the point 2 marked. The graph of the second inequality is a number line with the point 4 marked.
- Find the intersection of the inequalities: The intersection of the two inequalities is the region where the two number lines overlap.
The intersection of the two inequalities is the region between -∞ and 2, and the region between 4 and ∞.
- Write the solution set: The solution set is the region between -∞ and 2, and the region between 4 and ∞, which can be written as: x < 2 or x > 4
Which Set of Numbers is Included in the Solution Set of the Compound Inequality?
Now that we have learned how to solve compound inequalities, let's consider the following problem:
Which set of numbers is included in the solution set of the compound inequality?
A. {{-7, 5, 18, 24, 32}$}$ B. {{-9, 7, 15, 22, 26}$}$ C. {{16, 17, 22, 23, 24}$}$ D. {{18, 19, 20, 21, 22}$}$
To solve this problem, we need to determine which set of numbers is included in the solution set of the compound inequality.
Let's consider the compound inequality: 2 < x < 5 and x > 3
The solution set of this inequality is the region between 3 and 5, inclusive, which can be written as: 3 ≤ x ≤ 5
Now, let's examine each of the answer choices:
A. {{-7, 5, 18, 24, 32}$}$
This set of numbers includes the number 5, which is within the solution set of the compound inequality.
B. {{-9, 7, 15, 22, 26}$}$
This set of numbers does not include any numbers within the solution set of the compound inequality.
C. {{16, 17, 22, 23, 24}$}$
This set of numbers includes the number 22, which is within the solution set of the compound inequality.
D. {{18, 19, 20, 21, 22}$}$
This set of numbers includes the number 22, which is within the solution set of the compound inequality.
Based on the analysis above, we can conclude that the correct answer is:
A. {{-7, 5, 18, 24, 32}$}$
This set of numbers includes the number 5, which is within the solution set of the compound inequality.
Conclusion
Frequently Asked Questions
Q: What is a compound inequality?
A: A compound inequality is a mathematical expression that involves two or more inequalities joined by the words "and" or "or."
Q: How do I solve a compound inequality?
A: To solve a compound inequality, you need to follow these steps:
- Simplify the individual inequalities: Simplify each individual inequality by combining like terms and eliminating any unnecessary variables.
- Determine the relationship between the inequalities: Determine the relationship between the two inequalities, such as whether they are joined by "and" or "or."
- Graph the inequalities: Graph each individual inequality on a number line or coordinate plane.
- Find the intersection of the inequalities: Find the intersection of the two inequalities, which represents the solution set.
- Write the solution set: Write the solution set in interval notation or as a list of numbers.
Q: What is the difference between a conjunction and a disjunction compound inequality?
A: A conjunction compound inequality involves two or more inequalities joined by the word "and." A disjunction compound inequality involves two or more inequalities joined by the word "or."
Q: How do I graph a compound inequality?
A: To graph a compound inequality, you need to graph each individual inequality on a number line or coordinate plane. Then, find the intersection of the two inequalities, which represents the solution set.
Q: What is the solution set of a compound inequality?
A: The solution set of a compound inequality is the region where the two inequalities overlap. It can be written in interval notation or as a list of numbers.
Q: Can a compound inequality have more than two inequalities?
A: Yes, a compound inequality can have more than two inequalities. However, the process of solving it remains the same.
Q: How do I determine the relationship between the inequalities in a compound inequality?
A: To determine the relationship between the inequalities in a compound inequality, you need to look at the word that joins them. If the word is "and," the inequalities are joined by a conjunction. If the word is "or," the inequalities are joined by a disjunction.
Q: Can a compound inequality have a negative solution set?
A: Yes, a compound inequality can have a negative solution set. For example, the compound inequality x < -2 or x > 3 has a negative solution set.
Q: How do I write the solution set of a compound inequality in interval notation?
A: To write the solution set of a compound inequality in interval notation, you need to use the following notation:
- Union: The union of two or more intervals is denoted by the symbol ∪. For example, (2, 5) ∪ (7, 10) represents the union of the intervals (2, 5) and (7, 10).
- Intersection: The intersection of two or more intervals is denoted by the symbol ∩. For example, (2, 5) ∩ (7, 10) represents the intersection of the intervals (2, 5) and (7, 10).
Q: Can a compound inequality have a solution set that is a single number?
A: Yes, a compound inequality can have a solution set that is a single number. For example, the compound inequality x = 2 has a solution set that is a single number, 2.
Conclusion
In this article, we have answered some of the most frequently asked questions about compound inequalities. We have covered topics such as the definition of a compound inequality, how to solve a compound inequality, and how to write the solution set in interval notation. We hope that this article has been helpful in clarifying any confusion you may have had about compound inequalities.