Isolate the absolute value expression

Oct 13, 2025 by windailys.com 114 views

Introduction

In this article, we will delve into the world of absolute value inequalities and explore how to solve them. Absolute value inequalities involve absolute value expressions, which are expressions enclosed within absolute value symbols (| |). These inequalities can be challenging to solve, but with the right approach and techniques, we can break them down and find the solution.

What is an Absolute Value Inequality?

An absolute value inequality is an inequality that involves an absolute value expression. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5. Absolute value inequalities can be written in the form |x| > a, |x| < a, |x| ≥ a, or |x| ≤ a, where x is the variable and a is a constant.

Solving Absolute Value Inequalities: A Step-by-Step Guide

To solve an absolute value inequality, we need to follow these steps:

Write the inequality in the form |x| > a, |x| < a, |x| ≥ a, or |x| ≤ a. This will help us identify the type of inequality we are dealing with.
Isolate the absolute value expression. This means that we need to get the absolute value expression by itself on one side of the inequality.
Split the inequality into two separate inequalities. This is because the absolute value expression can be either positive or negative.
Solve each inequality separately. This means that we need to solve the inequality for both the positive and negative cases.
Combine the solutions. This means that we need to combine the solutions from both the positive and negative cases.

Solving the Inequality |1 - 4x| > 10

Now, let's apply these steps to solve the inequality |1 - 4x| > 10.

Step 1: Write the inequality in the form |x| > a

The given inequality is |1 - 4x| > 10. This is already in the form |x| > a, where x = 1 - 4x and a = 10.

Step 2: Isolate the absolute value expression

We can isolate the absolute value expression by subtracting 1 from both sides of the inequality:

|1 - 4x - 1| > 10 - 1

This simplifies to:

|-4x| > 9

Step 3: Split the inequality into two separate inequalities

We can split the inequality into two separate inequalities:

-4x > 9 or -4x < -9

Step 4: Solve each inequality separately

Let's solve each inequality separately:

Case 1: -4x > 9

To solve this inequality, we can divide both sides by -4:

x < -9/4

This simplifies to:

x < -2.25

Case 2: -4x < -9

To solve this inequality, we can divide both sides by -4:

x > -9/4

This simplifies to:

x > -2.25

Step 5: Combine the solutions

Since the two inequalities are mutually exclusive, we can combine the solutions by taking the union of the two solutions:

x < -2.25 or x > -2.25

This simplifies to:

x ≠ -2.25

Conclusion

In this article, we have learned how to solve absolute value inequalities using a step-by-step approach. We have applied this approach to solve the inequality |1 - 4x| > 10 and have found the solution to be x ≠ -2.25. We hope that this article has provided you with a clear understanding of how to solve absolute value inequalities and has given you the confidence to tackle more complex inequalities in the future.

Additional Resources

If you are struggling to understand absolute value inequalities or need additional practice, we recommend the following resources:

Khan Academy: Absolute Value Inequalities
Mathway: Absolute Value Inequalities
Wolfram Alpha: Absolute Value Inequalities

Frequently Asked Questions

Q: What is an absolute value inequality? A: An absolute value inequality is an inequality that involves an absolute value expression.

Q: How do I solve an absolute value inequality? A: To solve an absolute value inequality, you need to follow the steps outlined in this article.

Q: What is the solution to the inequality |1 - 4x| > 10? A: The solution to the inequality |1 - 4x| > 10 is x ≠ -2.25.

Introduction

In our previous article, we explored the world of absolute value inequalities and learned how to solve them using a step-by-step approach. However, we know that practice makes perfect, and there's no better way to practice than by answering questions and solving problems. In this article, we'll provide a Q&A section where we'll answer some of the most frequently asked questions about absolute value inequalities.

Q&A

Q: What is an absolute value inequality?

A: An absolute value inequality is an inequality that involves an absolute value expression. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to follow these steps:

Write the inequality in the form |x| > a, |x| < a, |x| ≥ a, or |x| ≤ a.
Isolate the absolute value expression.
Split the inequality into two separate inequalities.
Solve each inequality separately.
Combine the solutions.

Q: What is the difference between an absolute value inequality and a linear inequality?

A: An absolute value inequality involves an absolute value expression, while a linear inequality does not. For example, the inequality |x| > 2 is an absolute value inequality, while the inequality x > 2 is a linear inequality.

Q: Can I use a calculator to solve absolute value inequalities?

A: Yes, you can use a calculator to solve absolute value inequalities, but it is not always necessary. In some cases, a calculator can be a useful tool, but in other cases, it may not be necessary.

Q: How do I know which inequality to use when solving an absolute value inequality?

A: When solving an absolute value inequality, you need to use the correct inequality based on the given inequality. For example, if the given inequality is |x| > 2, you would use the inequality x > 2 or x < -2.

Q: What is the solution to the inequality |x| > 3?

A: The solution to the inequality |x| > 3 is x < -3 or x > 3.

Q: Can I use the same steps to solve a quadratic inequality as I would to solve an absolute value inequality?

A: No, you cannot use the same steps to solve a quadratic inequality as you would to solve an absolute value inequality. Quadratic inequalities involve quadratic expressions, while absolute value inequalities involve absolute value expressions.

Q: How do I know if an inequality is an absolute value inequality or a linear inequality?

A: To determine if an inequality is an absolute value inequality or a linear inequality, look for the absolute value symbol (| |). If the inequality involves an absolute value expression, it is an absolute value inequality. If the inequality does not involve an absolute value expression, it is a linear inequality.

Practice Problems

Problem 1: Solve the inequality |x| > 5

Problem 2: Solve the inequality |x| < 2

Problem 3: Solve the inequality |x| ≥ 3

Problem 4: Solve the inequality |x| ≤ 4

Solutions

Problem 1: Solve the inequality |x| > 5

The solution to the inequality |x| > 5 is x < -5 or x > 5.

Problem 2: Solve the inequality |x| < 2

The solution to the inequality |x| < 2 is -2 < x < 2.

Problem 3: Solve the inequality |x| ≥ 3

The solution to the inequality |x| ≥ 3 is x ≤ -3 or x ≥ 3.

Problem 4: Solve the inequality |x| ≤ 4

The solution to the inequality |x| ≤ 4 is -4 ≤ x ≤ 4.

Conclusion

In this article, we've provided a Q&A section where we've answered some of the most frequently asked questions about absolute value inequalities. We've also provided practice problems and solutions to help you practice and reinforce your understanding of absolute value inequalities. Remember, practice makes perfect, so be sure to practice solving absolute value inequalities regularly to become proficient in this area of mathematics.