The Basics of Absolute Value Inequalities

Oct 13, 2025 by miodeya.com 39 views

Understanding Compound Inequalities

Compound inequalities are a combination of two or more inequalities that are connected by logical operators such as "and" or "or". In this article, we will focus on solving compound inequalities of the form $|ax+b|>c$ , where $a$ , $b$ , and $c$ are constants. We will use the inequality $|3x+2|>7$ as an example to demonstrate how to solve compound inequalities.

The Basics of Absolute Value Inequalities

Before we dive into solving compound inequalities, let's review the basics of absolute value inequalities. An absolute value inequality is an inequality that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of $-3$ is $3$ , because $-3$ is $3$ units away from zero on the number line.

Solving Absolute Value Inequalities

To solve an absolute value inequality, we need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression is negative. Let's consider the inequality $|3x+2|>7$ . We can start by writing two separate inequalities:

$3x+2>7$
$3x+2<-7$

Solving the First Inequality

To solve the first inequality, we can subtract $2$ from both sides to get:

$3x>5$

Next, we can divide both sides by $3$ to get:

$x>\frac{5}{3}$

Solving the Second Inequality

To solve the second inequality, we can add $7$ to both sides to get:

$3x>-9$

Next, we can divide both sides by $3$ to get:

$x>-3$

Combining the Two Inequalities

Now that we have solved the two inequalities, we can combine them to get the compound inequality:

$x>\frac{5}{3}$ or $x>-3$

Which Compound Inequality Can Be Used to Solve the Inequality?

Now that we have solved the compound inequality, we can compare it to the options given in the problem. The correct compound inequality is:

$-7 \ \textless \ 3x+2 \ \textgreater \ 7$

This is because the compound inequality $-7 \ \textless \ 3x+2 \ \textgreater \ 7$ is equivalent to the compound inequality $x>\frac{5}{3}$ or $x>-3$ .

Conclusion

In this article, we have demonstrated how to solve compound inequalities of the form $|ax+b|>c$ . We have used the inequality $|3x+2|>7$ as an example to show how to solve compound inequalities. We have also compared the solution to the options given in the problem to determine which compound inequality can be used to solve the inequality.

Final Answer

The final answer is:

$-7 \ \textless \ 3x+2 \ \textgreater \ 7$

This is the correct compound inequality that can be used to solve the inequality $|3x+2|>7$ .

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities that are connected by logical operators such as "and" or "or". In this article, we will focus on solving compound inequalities of the form $|ax+b|>c$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression is negative. You can start by writing two separate inequalities and then solving each one separately.

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

A: A single inequality is a statement that compares two expressions, such as $x>5$ . A compound inequality, on the other hand, is a combination of two or more inequalities that are connected by logical operators such as "and" or "or".

Q: How do I combine two inequalities to get a compound inequality?

A: To combine two inequalities, you need to use the logical operators "and" or "or". For example, if you have two inequalities $x>5$ and $x<10$ , you can combine them to get the compound inequality $5<x<10$ .

Q: What is the correct order of operations when solving a compound inequality?

A: When solving a compound inequality, you need to follow the order of operations:

Solve the absolute value inequality by considering two cases: one where the expression inside the absolute value is positive, and one where the expression is negative.
Write two separate inequalities for each case.
Solve each inequality separately.
Combine the two inequalities to get the compound inequality.

Q: How do I determine which compound inequality is correct?

A: To determine which compound inequality is correct, you need to compare the solution to the options given in the problem. The correct compound inequality is the one that is equivalent to the solution.

Q: What are some common mistakes to avoid when solving compound inequalities?

A: Some common mistakes to avoid when solving compound inequalities include:

Not considering both cases when solving an absolute value inequality.
Not combining the two inequalities correctly.
Not following the order of operations.
Not comparing the solution to the options given in the problem.

Q: How can I practice solving compound inequalities?

A: You can practice solving compound inequalities by working through examples and exercises. You can also try solving compound inequalities on your own and then check your answers with a calculator or a teacher.

Q: What are some real-world applications of compound inequalities?

A: Compound inequalities have many real-world applications, including:

Finance: Compound inequalities can be used to calculate interest rates and investment returns.
Science: Compound inequalities can be used to model population growth and decay.
Engineering: Compound inequalities can be used to design and optimize systems.

Q: How can I use compound inequalities in my daily life?

A: You can use compound inequalities in your daily life by applying them to real-world problems. For example, you can use compound inequalities to calculate the cost of a product or to determine the best investment strategy.

Conclusion

In this article, we have answered some frequently asked questions about solving compound inequalities. We have covered topics such as the definition of a compound inequality, how to solve an absolute value inequality, and how to combine two inequalities to get a compound inequality. We have also discussed some common mistakes to avoid and how to practice solving compound inequalities.