Solve The Inequality For $x$ And Identify The Graph Of Its Solution.$|x+3|\ \textless \ 2$Choose The Answer That Gives Both The Correct Solution And The Correct Graph.A. Solution: $-5 \ \textless \ X \ \textless \ -1$B.

Introduction

Absolute value inequalities are a fundamental concept in algebra and mathematics. They involve solving equations and inequalities that contain absolute value expressions. In this article, we will focus on solving the inequality $|x+3| < 2$ and identifying the graph of its solution.

Understanding Absolute Value

Before we dive into solving the inequality, let's briefly review the concept of absolute value. The absolute value of a number $x$, denoted by $|x|$, is the distance of $x$ from zero on the number line. It is always non-negative, and it can be thought of as the magnitude or size of $x$. For example, the absolute value of $-3$ is $3$, and the absolute value of $4$ is also $4$.

Solving the Inequality

Now, let's solve the inequality $|x+3| < 2$. To do this, we need to consider two cases:

Case 1: $x+3 \geq 0$

In this case, the absolute value expression $|x+3|$ simplifies to $x+3$. We can then rewrite the inequality as:

$x+3 < 2$

Subtracting $3$ from both sides gives us:

$x < -1$

Case 2: $x+3 < 0$

In this case, the absolute value expression $|x+3|$ simplifies to $-(x+3)$. We can then rewrite the inequality as:

$-(x+3) < 2$

Multiplying both sides by $-1$ gives us:

$x+3 > -2$

Subtracting $3$ from both sides gives us:

$x > -5$

Combining the Cases

We have found two cases: $x < -1$ and $x > -5$. However, we need to combine these cases to get the final solution. Since the two cases are mutually exclusive (i.e., they cannot occur at the same time), we can use the union symbol $\cup$ to combine them:

$-5 < x < -1$

Graphing the Solution

To graph the solution, we need to plot the points $x=-5$ and $x=-1$ on the number line. We can then shade the region between these two points to represent the solution set.

Conclusion

In this article, we solved the inequality $|x+3| < 2$ and identified the graph of its solution. We used two cases to solve the inequality and then combined the cases to get the final solution. We also graphed the solution using the number line. The final answer is:

A. Solution: $-5 < x < -1$

Introduction

In our previous article, we solved the inequality $|x+3| < 2$ and identified the graph of its solution. However, we know that there are many more absolute value inequalities to solve, and each one requires a different approach. In this article, we will answer some frequently asked questions about solving absolute value inequalities.

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

A: An absolute value equation is an equation that contains an absolute value expression, such as $|x| = 5$. An absolute value inequality, on the other hand, is an inequality that contains an absolute value expression, such as $|x| < 5$.

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 it is negative. You then solve each case separately and combine the solutions.

Q: What is the union symbol in absolute value inequalities?

A: The union symbol, denoted by $\cup$, is used to combine two or more cases in an absolute value inequality. For example, if we have two cases $x < -1$ and $x > -5$, we can combine them using the union symbol as $-5 < x < -1$.

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

A: To graph the solution to an absolute value inequality, you need to plot the points that define the solution set on the number line. You can then shade the region between these points to represent the solution set.

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 solving an absolute value inequality
• Not combining the cases correctly using the union symbol
• Not graphing the solution correctly
• Not checking the solution for extraneous solutions

Q: How do I check for extraneous solutions in an absolute value inequality?

A: To check for extraneous solutions in an absolute value inequality, you need to plug the solution back into the original inequality and check if it is true. If it is not true, then the solution is extraneous and should be discarded.

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

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

• Modeling real-world problems that involve distances or magnitudes
• Solving problems that involve inequalities with absolute value expressions
• Graphing functions that involve absolute value expressions

Conclusion

In this article, we answered some frequently asked questions about solving absolute value inequalities. We covered topics such as the difference between absolute value equations and inequalities, how to solve absolute value inequalities, and how to graph the solution. We also discussed common mistakes to avoid and how to check for extraneous solutions. By following these tips and techniques, you can become proficient in solving absolute value inequalities and apply them to real-world problems.

Additional Resources

For more information on solving absolute value inequalities, check out the following resources:

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

Practice Problems

Try solving the following absolute value inequalities:

1. $|x-2| < 3$
2. $|x+1| > 4$
3. $|x-5| \leq 2$

Answer Key

1. $-1 < x < 5$
2. $x < -5$ or $x > 1$
3. $3 \leq x \leq 7$