Which Of The Following Is Equivalent To $|x-1|\ \textless \ 5$?A. $x-1\ \textless \ 5$ B. $5\ \textless \ X-1\ \textless \ 5$ C. $-5\ \textless \ X-1\ \textless \ 5$ D. $-5\ \textgreater \ X-1\

Introduction

Absolute value inequalities are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, calculus, and engineering. In this article, we will explore the concept of absolute value inequalities and provide a detailed explanation of the given problem: Which of the following is equivalent to $|x-1|\ \textless \ 5$?

What is an Absolute Value Inequality?

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, without considering direction. In other words, the absolute value of a number is always non-negative.

Understanding the Given Problem

The given problem is $|x-1|\ \textless \ 5$. This means that the distance between $x$ and $1$ is less than $5$. To solve this problem, we need to consider two cases: when $x-1$ is positive and when $x-1$ is negative.

Case 1: When $x-1$ is Positive

When $x-1$ is positive, the absolute value of $x-1$ is equal to $x-1$. Therefore, the inequality becomes $x-1\ \textless \ 5$. Solving this inequality, we get $x\ \textless \ 6$.

Case 2: When $x-1$ is Negative

When $x-1$ is negative, the absolute value of $x-1$ is equal to $-(x-1)$. Therefore, the inequality becomes $-(x-1)\ \textless \ 5$. Simplifying this inequality, we get $-x+1\ \textless \ 5$, which further simplifies to $-x\ \textless \ 4$. Multiplying both sides by $-1$, we get $x\ \textgreater \ -4$.

Combining the Two Cases

Combining the two cases, we get $x\ \textless \ 6$ and $x\ \textgreater \ -4$. However, this is not the only solution. We also need to consider the values of $x$ that make $x-1$ equal to $5$ or $-5$. These values are $x=6$ and $x=-4$, respectively.

Conclusion

In conclusion, the given problem $|x-1|\ \textless \ 5$ is equivalent to $-5\ \textless \ x-1\ \textless \ 5$. This means that the distance between $x$ and $1$ is less than $5$, and the values of $x$ that satisfy this inequality are $x\ \textless \ 6$ and $x\ \textgreater \ -4$.

Answer

The correct answer is C. $-5\ \textless \ x-1\ \textless \ 5$.

Why is this the Correct Answer?

This is the correct answer because it accurately represents the solution to the given problem. The inequality $-5\ \textless \ x-1\ \textless \ 5$ means that the distance between $x$ and $1$ is less than $5$, which is equivalent to the original problem $|x-1|\ \textless \ 5$.

What are the Other Options?

The other options are:

A. $x-1\ \textless \ 5$

This option is incorrect because it only considers the case when $x-1$ is positive, and it does not account for the values of $x$ that make $x-1$ equal to $-5$.

B. $5\ \textless \ x-1\ \textless \ 5$

This option is incorrect because it is a contradictory statement. The inequality $5\ \textless \ x-1\ \textless \ 5$ means that $x-1$ is both greater than $5$ and less than $5$, which is impossible.

D. $-5\ \textgreater \ x-1$

This option is incorrect because it only considers the case when $x-1$ is negative, and it does not account for the values of $x$ that make $x-1$ equal to $5$.

Conclusion

In conclusion, the correct answer is C. $-5\ \textless \ x-1\ \textless \ 5$. This option accurately represents the solution to the given problem, and it is the only option that correctly accounts for all the possible values of $x$ that satisfy the inequality $|x-1|\ \textless \ 5$.

Final Thoughts

Absolute value inequalities are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, calculus, and engineering. In this article, we have explored the concept of absolute value inequalities and provided a detailed explanation of the given problem. We have also discussed the other options and explained why they are incorrect. We hope that this article has provided a clear understanding of the concept of absolute value inequalities and has helped readers to better understand the given problem.

Introduction

In our previous article, we explored the concept of absolute value inequalities and provided a detailed explanation of the given problem: Which of the following is equivalent to $|x-1|\ \textless \ 5$? In this article, we will continue to discuss absolute value inequalities and provide a Q&A section to help readers better understand the concept.

Q&A Section

Q: What is an absolute value inequality?

A: 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, without considering direction.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to consider two cases: when the expression inside the absolute value is positive and when it is negative. You can then use the properties of absolute value to simplify the inequality and find the solution.

Q: What are the properties of absolute value?

A: The properties of absolute value are:

• The absolute value of a number is always non-negative.
• The absolute value of a number is equal to the number if it is positive, and equal to the negative of the number if it is negative.
• The absolute value of a product or quotient is equal to the product or quotient of the absolute values.

Q: How do I simplify an absolute value inequality?

A: To simplify an absolute value inequality, you can use the properties of absolute value to rewrite the inequality in a simpler form. For example, if you have the inequality $|x-1|\ \textless \ 5$, you can rewrite it as $-5\ \textless \ x-1\ \textless \ 5$.

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

A: Some common mistakes to avoid when solving absolute value inequalities include:

• Not considering both cases when the expression inside the absolute value is positive and negative.
• Not using the properties of absolute value to simplify the inequality.
• Not checking the solution to make sure it satisfies the original inequality.

Q: How do I check the solution to an absolute value inequality?

A: To check the solution to an absolute value inequality, you can plug the solution back into the original inequality and make sure it is true. You can also use a number line or a graph to visualize the solution and make sure it is correct.

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

A: Absolute value inequalities have many real-world applications, including:

• Physics: Absolute value inequalities are used to describe the motion of objects and the forces acting on them.
• Engineering: Absolute value inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Absolute value inequalities are used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, absolute value inequalities are a fundamental concept in mathematics, and they have many real-world applications. In this article, we have provided a Q&A section to help readers better understand the concept and solve absolute value inequalities. We hope that this article has been helpful and has provided a clear understanding of the concept of absolute value inequalities.

Final Thoughts

Absolute value inequalities are a powerful tool for solving problems in mathematics and other fields. By understanding the properties of absolute value and how to simplify absolute value inequalities, you can solve a wide range of problems and make predictions about future trends. We hope that this article has been helpful and has provided a clear understanding of the concept of absolute value inequalities.