Verifying the Solution Set

Oct 13, 2025 by windailys.com 100 views

Introduction

Absolute value inequalities are a fundamental concept in mathematics, particularly in algebra and calculus. They involve the absolute value of a variable or expression, which is always non-negative. In this article, we will explore the solution set to the inequality $|x+3| < 5$ and verify which statements confirm that the solution set is $-8 < x < 2$ .

The Inequality $|x+3| < 5$

The given inequality is $|x+3| < 5$ . To solve this inequality, we need to consider two cases:

Case 1: $x+3 \geq 0$
Case 2: $x+3 < 0$

Case 1: $x+3 \geq 0$

In this case, the absolute value inequality becomes $x+3 < 5$ . To solve for $x$ , we subtract 3 from both sides:

x+3-3 < 5-3

x < 2

Case 2: $x+3 < 0$

In this case, the absolute value inequality becomes $-(x+3) < 5$ . To solve for $x$ , we multiply both sides by -1 and flip the inequality sign:

-(x+3) < 5

-x-3 < 5

-x < 5+3

-x < 8

x > -8

Combining the Cases

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

-8 < x < 2

Verifying the Solution Set

Now that we have the solution set $-8 < x < 2$ , let's verify which statements confirm this solution set.

Substituting a Value into the Inequality from the Solution Set

Statement 1: Substituting a value into the inequality from the solution set, such as $-2$ , will always satisfy the inequality.

Verification: Let's substitute $x = -2$ into the inequality $|x+3| < 5$ :

|(-2)+3| < 5

|1| < 5

1 < 5

Since $1 < 5$ is true, the statement is verified.

Checking the Endpoints of the Solution Set

Statement 2: The endpoints of the solution set, $x = -8$ and $x = 2$ , are not included in the solution set.

Verification: Let's check the endpoint $x = -8$ :

|(-8)+3| < 5

|-5| < 5

5 < 5

Since $5 < 5$ is false, the endpoint $x = -8$ is not included in the solution set.

Let's check the endpoint $x = 2$ :

|(2)+3| < 5

|5| < 5

5 < 5

Since $5 < 5$ is false, the endpoint $x = 2$ is not included in the solution set.

Checking the Intervals within the Solution Set

Statement 3: The intervals within the solution set, $-8 < x < -5$ and $-5 < x < 2$ , are included in the solution set.

Verification: Let's check the interval $-8 < x < -5$ :

For $x = -6$ :

|(-6)+3| < 5

|-3| < 5

3 < 5

Since $3 < 5$ is true, the interval $-8 < x < -5$ is included in the solution set.

Let's check the interval $-5 < x < 2$ :

For $x = 0$ :

|(0)+3| < 5

|3| < 5

3 < 5

Since $3 < 5$ is true, the interval $-5 < x < 2$ is included in the solution set.

Conclusion

In conclusion, the statements that verify that the solution set to $|x+3| < 5$ is $-8 < x < 2$ are:

Substituting a value into the inequality from the solution set, such as $-2$ , will always satisfy the inequality.
The endpoints of the solution set, $x = -8$ and $x = 2$ , are not included in the solution set.
The intervals within the solution set, $-8 < x < -5$ and $-5 < x < 2$ , are included in the solution set.

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:

Case 1: The expression inside the absolute value is non-negative.
Case 2: The expression inside the absolute value is negative.

You then solve each case separately and combine the solutions to find the final solution set.

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 that an absolute value inequality involves the absolute value of an expression, which is always non-negative. This means that the solution set to an absolute value inequality may include both positive and negative values.

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

A: No, you cannot use the same methods 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 non-negative, you use Case 1. If the expression is negative, you use Case 2.

Q: Can I use a calculator to solve absolute value inequalities?

A: Yes, you can use a calculator to solve absolute value inequalities. However, you need to be careful when using a calculator, as it may not always give you the correct solution set.

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

A: To check your solution set, you need to substitute a value from the solution set into the original inequality and verify that it is true. You can also use a calculator to check your 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 the inequality.
Not combining the solutions from both cases to find the final solution set.
Not checking the solution set to ensure that it is correct.

Q: How do I graph an absolute value inequality on a number line?

A: To graph an absolute value inequality on a number line, you need to plot the solution set on the number line and use a closed circle to indicate the endpoints of the solution set.

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

A: Yes, you can use absolute value inequalities to solve real-world problems. Absolute value inequalities are commonly used in physics, engineering, and other fields to model real-world problems.

Conclusion

In conclusion, absolute value inequalities are a fundamental concept in mathematics that involve the absolute value of a variable or expression. They require a different approach than regular inequalities and can be used to solve real-world problems. By understanding how to solve absolute value inequalities, you can apply this knowledge to a wide range of fields and problems.