Which Compound Inequalities Have \[$ X = 2 \$\] As A Solution? Check All That Apply.1. \[$ 4 \ \textless \ 5x - 1 \ \textless \ 10 \$\]2. \[$ 4 \ \textless \ 5x - 3 \ \textless \ 10 \$\]3. \[$ 4 \ \textless \ 5x -
Introduction
In mathematics, compound inequalities are a type of inequality that involves multiple expressions or conditions. They are used to describe a range of values that a variable can take, and are often used in algebra and calculus. In this article, we will explore compound inequalities and determine which ones have a specific solution, x = 2.
What are Compound Inequalities?
Compound inequalities are inequalities that involve multiple expressions or conditions. They can be written in the form:
a < f(x) < b
or
a > f(x) > b
where a and b are constants, and f(x) is a function of x.
Types of Compound Inequalities
There are two main types of compound inequalities:
- Double inequality: This type of inequality involves two expressions or conditions, and is written in the form:
a < f(x) < b
or
a > f(x) > b
- Compound inequality with absolute value: This type of inequality involves an absolute value expression, and is written in the form:
|f(x)| < a
or
|f(x)| > a
Solving Compound Inequalities
To solve a compound inequality, we need to find the values of x that satisfy both expressions or conditions. This can be done by solving each expression or condition separately, and then combining the solutions.
Example 1: Solving a Double Inequality
Let's consider the compound inequality:
4 < 5x - 1 < 10
To solve this inequality, we need to find the values of x that satisfy both expressions:
5x - 1 > 4
and
5x - 1 < 10
Solving the first expression, we get:
5x > 5
x > 1
Solving the second expression, we get:
5x < 11
x < 2.2
Combining the solutions, we get:
1 < x < 2.2
Example 2: Solving a Compound Inequality with Absolute Value
Let's consider the compound inequality:
|5x - 3| < 4
To solve this inequality, we need to find the values of x that satisfy the absolute value expression:
-4 < 5x - 3 < 4
Solving the first expression, we get:
5x - 3 > -4
5x > -1
x > -0.2
Solving the second expression, we get:
5x - 3 < 4
5x < 7
x < 1.4
Combining the solutions, we get:
-0.2 < x < 1.4
Which Compound Inequalities Have x = 2 as a Solution?
Now that we have a better understanding of compound inequalities and how to solve them, let's determine which ones have x = 2 as a solution.
- 4 < 5x - 1 < 10
To determine if x = 2 is a solution, we need to plug in x = 2 into the inequality:
4 < 5(2) - 1 < 10
Simplifying the expression, we get:
4 < 9 < 10
Since 9 is indeed between 4 and 10, x = 2 is a solution to this inequality.
- 4 < 5x - 3 < 10
To determine if x = 2 is a solution, we need to plug in x = 2 into the inequality:
4 < 5(2) - 3 < 10
Simplifying the expression, we get:
4 < 7 < 10
Since 7 is indeed between 4 and 10, x = 2 is a solution to this inequality.
- 4 < 5x - 4 < 10
To determine if x = 2 is a solution, we need to plug in x = 2 into the inequality:
4 < 5(2) - 4 < 10
Simplifying the expression, we get:
4 < 6 < 10
Since 6 is indeed between 4 and 10, x = 2 is a solution to this inequality.
Conclusion
In this article, we explored compound inequalities and determined which ones have x = 2 as a solution. We saw that compound inequalities can be used to describe a range of values that a variable can take, and are often used in algebra and calculus. We also saw that x = 2 is a solution to the compound inequalities:
- 4 < 5x - 1 < 10
- 4 < 5x - 3 < 10
- 4 < 5x - 4 < 10
Q: What is a compound inequality?
A: A compound inequality is a type of inequality that involves multiple expressions or conditions. It can be written in the form:
a < f(x) < b
or
a > f(x) > b
where a and b are constants, and f(x) is a function of x.
Q: How do I solve a compound inequality?
A: To solve a compound inequality, you need to find the values of x that satisfy both expressions or conditions. This can be done by solving each expression or condition separately, and then combining the solutions.
Q: What are the different types of compound inequalities?
A: There are two main types of compound inequalities:
- Double inequality: This type of inequality involves two expressions or conditions, and is written in the form:
a < f(x) < b
or
a > f(x) > b
- Compound inequality with absolute value: This type of inequality involves an absolute value expression, and is written in the form:
|f(x)| < a
or
|f(x)| > a
Q: How do I determine if a value is a solution to a compound inequality?
A: To determine if a value is a solution to a compound inequality, you need to plug the value into the inequality and check if it satisfies both expressions or conditions.
Q: Can a compound inequality have multiple solutions?
A: Yes, a compound inequality can have multiple solutions. This can happen when the inequality has multiple intervals or when the solution set is a union of multiple intervals.
Q: How do I graph a compound inequality?
A: To graph a compound inequality, you need to graph each expression or condition separately, and then combine the graphs. The solution set will be the intersection of the two graphs.
Q: Can a compound inequality be written in interval notation?
A: Yes, a compound inequality can be written in interval notation. For example, the compound inequality:
-2 < x < 3
can be written in interval notation as:
(-2, 3)
Q: How do I solve a compound inequality with absolute value?
A: To solve a compound inequality with absolute value, you need to isolate the absolute value expression and then solve for x. You can use the following steps:
- Isolate the absolute value expression.
- Set up two equations: one with the positive sign and one with the negative sign.
- Solve each equation separately.
- Combine the solutions to find the final answer.
Q: Can a compound inequality have a solution set that is a single point?
A: Yes, a compound inequality can have a solution set that is a single point. This can happen when the inequality has a single solution that satisfies both expressions or conditions.
Q: How do I determine if a compound inequality is true or false?
A: To determine if a compound inequality is true or false, you need to check if the solution set is empty or not. If the solution set is empty, then the inequality is false. If the solution set is not empty, then the inequality is true.
Q: Can a compound inequality be used to model real-world problems?
A: Yes, a compound inequality can be used to model real-world problems. For example, a compound inequality can be used to model the range of values that a variable can take in a given situation.
Q: How do I use a compound inequality to solve a real-world problem?
A: To use a compound inequality to solve a real-world problem, you need to:
- Identify the variables and constants in the problem.
- Write the compound inequality based on the problem.
- Solve the compound inequality to find the solution set.
- Interpret the solution set in the context of the problem.
Conclusion
In this article, we have answered some common questions about compound inequalities. We have discussed the different types of compound inequalities, how to solve them, and how to graph them. We have also discussed how to use compound inequalities to model real-world problems and how to solve them in the context of a problem.