What Is The Solution To The Inequality ∣ 2 N + 5 ∣ \textgreater 1 |2n + 5| \ \textgreater \ 1 ∣2 N + 5∣ \textgreater 1 ?A. − 3 \textgreater N \textgreater − 2 -3 \ \textgreater \ N \ \textgreater \ -2 − 3 \textgreater N \textgreater − 2 B. 2 \textless N \textless 3 2 \ \textless \ N \ \textless \ 3 2 \textless N \textless 3 C. N \textless − 3 N \ \textless \ -3 N \textless − 3 Or $n \

$$

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more values. The absolute value inequality is a type of inequality that involves the absolute value of an expression. In this article, we will focus on solving the absolute value inequality $|2n + 5| \ \textgreater \ 1$. We will break down the solution step by step and provide a clear explanation of each step.

Understanding Absolute Value Inequalities

Before we dive into solving the inequality, let's first 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 5 is 5, and the absolute value of -5 is also 5. When we have an absolute value inequality, we are looking for the values of the variable that make the absolute value expression greater than or less than a certain value.

Solving the Inequality

To solve the inequality $|2n + 5| \ \textgreater \ 1$, we need to consider two cases:

Case 1: $2n + 5 \geq 0$

When $2n + 5 \geq 0$, we can remove the absolute value sign and solve the inequality as follows:

$2n + 5 \ \textgreater \ 1$

Subtracting 5 from both sides gives us:

$2n \ \textgreater \ -4$

Dividing both sides by 2 gives us:

$n \ \textgreater \ -2$

Case 2: $2n + 5 \lt 0$

When $2n + 5 \lt 0$, we need to multiply both sides of the inequality by -1 to get rid of the negative sign. This will flip the direction of the inequality:

$-(2n + 5) \ \textgreater \ -1$

Distributing the negative sign gives us:

$-2n - 5 \ \textgreater \ -1$

Adding 5 to both sides gives us:

$-2n \ \textgreater \ 4$

Dividing both sides by -2 gives us:

$n \ \lt \ -2$

Combining the Solutions

Now that we have solved the inequality for both cases, we can combine the solutions to get the final answer. We have:

$n \ \textgreater \ -2$ (from Case 1)

and

$n \ \lt \ -2$ (from Case 2)

Since both inequalities are true, we can combine them to get:

$n \ \textless \ -2$

Conclusion

In this article, we solved the absolute value inequality $|2n + 5| \ \textgreater \ 1$. We broke down the solution into two cases and used algebraic manipulations to isolate the variable. The final answer is $n \ \textless \ -2$. This solution can be represented graphically on a number line, where all values less than -2 satisfy the inequality.

Frequently Asked Questions

• What is the absolute value inequality?
• How do we solve absolute value inequalities?
• What is the solution to the inequality $|2n + 5| \ \textgreater \ 1$?

Final Answer

The final answer is $n \ \textless \ -2$.

Introduction

In our previous article, we solved the absolute value inequality $|2n + 5| \ \textgreater \ 1$. We received many questions from readers who were struggling to understand the concept of absolute value inequalities. In this article, we will answer some of the most frequently asked questions about absolute value inequalities.

Q&A

Q: What is the absolute value inequality?

A: The absolute value inequality is a type of inequality that involves the absolute value of an expression. It is a mathematical statement that compares the absolute value of an expression to a certain value.

Q: How do we solve absolute value inequalities?

A: To solve absolute value inequalities, we need to consider two cases: when the expression inside the absolute value is non-negative and when it is negative. We then use algebraic manipulations to isolate the variable.

Q: What is the difference between absolute value inequalities and linear inequalities?

A: Absolute value inequalities involve the absolute value of an expression, while linear inequalities do not. Linear inequalities are simpler and can be solved using basic algebraic manipulations.

Q: Can we use the same method to solve all absolute value inequalities?

A: No, we cannot use the same method to solve all absolute value inequalities. The method we used in our previous article is specific to inequalities of the form $|ax + b| \ \textgreater \ c$. We need to adjust the method depending on the specific inequality we are solving.

Q: How do we represent absolute value inequalities graphically?

A: We can represent absolute value inequalities graphically on a number line. The solution to the inequality is the set of all values that lie outside the absolute value expression.

Q: Can we use absolute value inequalities to solve real-world problems?

A: Yes, we can use absolute value inequalities to solve real-world problems. For example, we can use absolute value inequalities to model the distance between two objects or the amount of money spent on a particular item.

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 non-negative and when it is negative)
• Not using the correct algebraic manipulations to isolate the variable
• Not checking the solution to ensure that it satisfies the original inequality

Conclusion

In this article, we answered some of the most frequently asked questions about absolute value inequalities. We hope that this article has helped to clarify any confusion and provide a better understanding of absolute value inequalities.

Additional Resources

• For more information on absolute value inequalities, please see our previous article on the topic.
• For practice problems and exercises, please see our online resources page.
• For additional resources and support, please contact us at [insert contact information].

Final Answer

The final answer is that absolute value inequalities are a powerful tool for solving mathematical problems and modeling real-world situations. By understanding how to solve absolute value inequalities, we can gain a deeper understanding of mathematical concepts and develop problem-solving skills that can be applied to a wide range of situations.