Solve The Inequality:${ |4x - 1| \ \textgreater \ 8 }$

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Introduction

In this article, we will delve into the world of inequalities and focus on solving the given inequality $|4x - 1| > 8$. Inequalities are mathematical expressions that compare two values, and they can be solved using various techniques. The absolute value inequality is a type of inequality that involves the absolute value of an expression. In this case, we have the absolute value of $4x - 1$ greater than $8$. Our goal is to find the values of $x$ that satisfy this inequality.

Understanding Absolute Value Inequalities

Before we dive into solving the inequality, let's understand what absolute value inequalities are. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of $-3$ is $3$, and the absolute value of $4$ is also $4$. When we have an absolute value inequality, we are looking for the values of the variable that make the absolute value expression greater than, less than, or equal to a certain value.

Solving the Inequality

To solve the inequality $|4x - 1| > 8$, we need to consider two cases:

Case 1: $4x - 1 > 8$

In this case, we can start by isolating the variable $x$. We can do this by adding $1$ to both sides of the inequality:

$$4x - 1 + 1 > 8 + 1$$

This simplifies to:

$$4x > 9$$

Next, we can divide both sides of the inequality by $4$ to solve for $x$:

$$\frac{4x}{4} > \frac{9}{4}$$

This gives us:

$$x > \frac{9}{4}$$

Case 2: $4x - 1 < -8$

In this case, we can start by isolating the variable $x$. We can do this by adding $1$ to both sides of the inequality:

$$4x - 1 + 1 < -8 + 1$$

This simplifies to:

$$4x < -7$$

Next, we can divide both sides of the inequality by $4$ to solve for $x$:

$$\frac{4x}{4} < \frac{-7}{4}$$

This gives us:

$$x < -\frac{7}{4}$$

Combining the Cases

Now that we have solved the two cases, we can combine them to find the final solution. We have:

$$x > \frac{9}{4} \text{ or } x < -\frac{7}{4}$$

This means that the solution to the inequality $|4x - 1| > 8$ is all real numbers $x$ that are greater than $\frac{9}{4}$ or less than $-\frac{7}{4}$.

Conclusion

In this article, we have solved the inequality $|4x - 1| > 8$. We have used the concept of absolute value inequalities and have considered two cases to find the final solution. The solution to the inequality is all real numbers $x$ that are greater than $\frac{9}{4}$ or less than $-\frac{7}{4}$. We hope that this article has provided a clear understanding of how to solve absolute value inequalities.

Frequently Asked Questions

• What is an absolute value inequality? An absolute value inequality is a type of inequality that involves the absolute value of an expression.
• How do I solve an absolute value inequality? 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 inside the absolute value is negative.
• What is the solution to the inequality $|4x - 1| > 8$? The solution to the inequality $|4x - 1| > 8$ is all real numbers $x$ that are greater than $\frac{9}{4}$ or less than $-\frac{7}{4}$.

Further Reading

• Absolute Value Inequalities: A Comprehensive Guide
• Solving Inequalities: A Step-by-Step Guide
• Algebra: A Beginner's Guide

References

• [1] "Algebra" by Michael Artin
• [2] "Inequalities" by Michael Spivak
• [3] "Absolute Value Inequalities" by James Stewart
  

Introduction

In our previous article, we solved the inequality $|4x - 1| > 8$ using the concept of absolute value inequalities. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have. We will cover a range of topics related to absolute value inequalities, from the basics to more advanced concepts.

Q&A

Q: What is an absolute value inequality?

A: An absolute value inequality is a type of inequality that involves the absolute value of an expression. It is used to compare the distance of the expression from zero on the number line.

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 inside the absolute value is negative. You can then use algebraic techniques to solve for the variable.

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

A: The main difference between an absolute value inequality and a regular inequality is the presence of the absolute value symbol. Absolute value inequalities involve the distance of the expression from zero, while regular inequalities involve a direct comparison.

Q: Can I use the same techniques to solve absolute value inequalities as I would for regular inequalities?

A: No, you cannot use the same techniques to solve absolute value inequalities as you would for regular inequalities. Absolute value inequalities require a different approach, as you need to consider the two cases mentioned earlier.

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

A: To determine which case to use, you need to consider the sign of the expression inside the absolute value. If the expression is positive, you use the first case. If the expression is negative, you use the second case.

Q: Can I use absolute value inequalities to solve equations?

A: Yes, you can use absolute value inequalities to solve equations. However, you need to be careful when using this technique, as it can lead to extraneous solutions.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not valid for the original equation. When using absolute value inequalities to solve equations, you need to check each solution to ensure that it is not extraneous.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug each solution back into the original equation and verify that it is true.

Q: Can I use absolute value inequalities to solve systems of equations?

A: Yes, you can use absolute value inequalities to solve systems of equations. However, you need to be careful when using this technique, as it can lead to extraneous solutions.

Q: What is the solution to the inequality $|4x - 1| > 8$?

A: The solution to the inequality $|4x - 1| > 8$ is all real numbers $x$ that are greater than $\frac{9}{4}$ or less than $-\frac{7}{4}$.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts or questions that readers may have about absolute value inequalities. We have covered a range of topics, from the basics to more advanced concepts. We hope that this article has provided a clear understanding of how to solve absolute value inequalities.

Frequently Asked Questions

• What is an absolute value inequality?
• How do I solve an absolute value inequality?
• What is the difference between an absolute value inequality and a regular inequality?
• Can I use the same techniques to solve absolute value inequalities as I would for regular inequalities?
• How do I know which case to use when solving an absolute value inequality?
• Can I use absolute value inequalities to solve equations?
• What is an extraneous solution?
• How do I check for extraneous solutions?
• Can I use absolute value inequalities to solve systems of equations?

Further Reading

• Absolute Value Inequalities: A Comprehensive Guide
• Solving Inequalities: A Step-by-Step Guide
• Algebra: A Beginner's Guide

References

• [1] "Algebra" by Michael Artin
• [2] "Inequalities" by Michael Spivak
• [3] "Absolute Value Inequalities" by James Stewart