Select The Correct Answer.Which Number Line Represents The Solution To $|x-5|\ \textless \ 3$?

Introduction

Absolute value inequalities are a fundamental concept in mathematics, and they play a crucial role in solving various problems in algebra, geometry, and other branches of mathematics. In this article, we will focus on understanding and solving absolute value inequalities, specifically the inequality $|x-5| < 3$. We will explore the concept of absolute value, learn how to solve absolute value inequalities, and determine which number line represents the solution to the given inequality.

What is Absolute Value?

Absolute value, denoted by the symbol $|x|$, is the distance of a number $x$ from zero on the number line. It is a measure of the magnitude of a number, without considering its direction. For example, the absolute value of $-3$ is $3$, and the absolute value of $3$ is also $3$. The absolute value of a number is always non-negative.

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 another where the expression is negative. Let's consider the inequality $|x-5| < 3$. We can rewrite this inequality as a double inequality:

$$-3 < x-5 < 3$$

Case 1: $x-5$ is Positive

When $x-5$ is positive, we can add $5$ to all three parts of the inequality:

$$-3+5 < x-5+5 < 3+5$$

Simplifying the inequality, we get:

$$2 < x < 8$$

Case 2: $x-5$ is Negative

When $x-5$ is negative, we can subtract $5$ from all three parts of the inequality:

$$-3-5 < x-5-5 < 3-5$$

Simplifying the inequality, we get:

$$-8 < x < -2$$

Combining the Cases

Since the absolute value inequality $|x-5| < 3$ is true for both cases, we can combine the solutions:

$$-8 < x < 8$$

Which Number Line Represents the Solution?

To determine which number line represents the solution to the inequality $|x-5| < 3$, we need to consider the solution set $-8 < x < 8$. This solution set represents all the numbers between $-8$ and $8$, excluding the endpoints.

Number Line Representation

The number line representation of the solution set $-8 < x < 8$ is a line segment that extends from $-8$ to $8$, excluding the endpoints. This line segment represents all the numbers that satisfy the inequality $|x-5| < 3$.

Conclusion

In conclusion, we have learned how to solve absolute value inequalities, specifically the inequality $|x-5| < 3$. We have explored the concept of absolute value, learned how to solve absolute value inequalities, and determined which number line represents the solution to the given inequality. The solution set $-8 < x < 8$ represents all the numbers that satisfy the inequality, and the number line representation is a line segment that extends from $-8$ to $8$, excluding the endpoints.

Key Takeaways

• Absolute value inequalities are a fundamental concept in mathematics.
• To solve an absolute value inequality, we need to consider two cases: one where the expression inside the absolute value is positive, and another where the expression is negative.
• The solution set of an absolute value inequality represents all the numbers that satisfy the inequality.
• The number line representation of a solution set is a line segment that extends from the lower endpoint to the upper endpoint, excluding the endpoints.

Final Thoughts

Introduction

In our previous article, we explored the concept of absolute value inequalities and learned how to solve them. In this article, we will provide a Q&A guide to help you better understand and apply the concepts of absolute value inequalities.

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. It is a mathematical statement that compares the absolute value of an expression to a constant or another expression.

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 another where the expression is negative. You can then use the properties of absolute value to simplify the inequality and find the solution set.

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

A: An absolute value inequality is a type of inequality that involves the absolute value of a variable or expression, while a linear inequality is a type of inequality that involves a linear expression. Absolute value inequalities are more complex and require a different approach to solve.

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

A: No, you cannot use the same methods to solve absolute value inequalities as you would for linear inequalities. Absolute value inequalities require a different approach, including considering two cases and using the properties of absolute value.

Q: How do I determine which number line represents the solution to an absolute value inequality?

A: To determine which number line represents the solution to an absolute value inequality, you need to consider the solution set and find the corresponding number line representation. This involves identifying the lower and upper endpoints of the solution set and drawing a line segment that extends from the lower endpoint to the upper endpoint, excluding the endpoints.

Q: Can I use absolute value inequalities to solve problems in real-world applications?

A: Yes, absolute value inequalities can be used to solve problems in real-world applications, such as finance, engineering, and physics. They can be used to model and solve problems involving distances, rates, and other quantities.

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:

• Failing to consider both cases (positive and negative)
• Not using the properties of absolute value correctly
• Not identifying the correct number line representation
• Not checking the solution set for extraneous solutions

Q: How can I practice solving absolute value inequalities?

A: You can practice solving absolute value inequalities by working through examples and exercises, such as those found in textbooks or online resources. You can also try solving problems that involve real-world applications to see how absolute value inequalities can be used to model and solve problems.

Conclusion

In conclusion, absolute value inequalities are a fundamental concept in mathematics that can be used to solve a wide range of problems. By understanding and applying the concepts of absolute value inequalities, you can develop a deeper understanding of mathematical concepts and improve your problem-solving skills. We hope this Q&A guide has been helpful in providing you with a better understanding of absolute value inequalities.

Key Takeaways

• Absolute value inequalities are a type of inequality that involves the absolute value of a variable or expression.
• To solve an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value is positive, and another where the expression is negative.
• The solution set of an absolute value inequality represents all the numbers that satisfy the inequality.
• The number line representation of a solution set is a line segment that extends from the lower endpoint to the upper endpoint, excluding the endpoints.
• Absolute value inequalities can be used to solve problems in real-world applications, such as finance, engineering, and physics.

Final Thoughts

Absolute value inequalities are an essential concept in mathematics that can be used to solve a wide range of problems. By understanding and applying the concepts of absolute value inequalities, you can develop a deeper understanding of mathematical concepts and improve your problem-solving skills. We hope this Q&A guide has been helpful in providing you with a better understanding of absolute value inequalities.