Split the inequality into two cases

Oct 13, 2025 by wbrum.org 131 views

Introduction

In this article, we will be solving the inequality $|x-4| \geq 9$ . This type of inequality is known as an absolute value inequality, and it involves the absolute value of a quantity. Our goal is to find the values of $x$ that satisfy this inequality and express the solution as an interval.

Understanding Absolute Value Inequalities

Before we dive into solving the inequality, let's take a moment to understand what absolute value inequalities are and how they work. The absolute value of a quantity $a$ is denoted by $|a|$ and is defined as the distance of $a$ from zero on the number line. In other words, it is the magnitude of $a$ without considering its direction.

For example, if we have $|x-4| \geq 9$ , this means that the distance of $x-4$ from zero is greater than or equal to 9. This can be visualized as a number line with the point 4 marked on it. Any point on the number line that is at least 9 units away from 4 will satisfy the inequality.

Solving the Inequality

Now that we have a good understanding of absolute value inequalities, let's move on to solving the inequality $|x-4| \geq 9$ . To solve this inequality, we will use the following steps:

Split the inequality into two cases: We will split the inequality into two separate inequalities, one for the positive case and one for the negative case.
Solve each case separately: We will solve each case separately using basic algebraic manipulations.
Combine the solutions: We will combine the solutions from each case to get the final solution.

Case 1: $x-4 \geq 9$

In this case, we have $x-4 \geq 9$ . To solve this inequality, we can add 4 to both sides, which gives us $x \geq 13$ .

Case 2: $x-4 \leq -9$

In this case, we have $x-4 \leq -9$ . To solve this inequality, we can add 4 to both sides, which gives us $x \leq -5$ .

Combining the Solutions

Now that we have solved each case separately, we can combine the solutions to get the final solution. The solution to the inequality $|x-4| \geq 9$ is the union of the two intervals $x \geq 13$ and $x \leq -5$ . This can be expressed as the interval $(-\infty, -5] \cup [13, \infty)$ .

Conclusion

In this article, we have solved the inequality $|x-4| \geq 9$ and expressed the solution as an interval. We have also taken a moment to understand what absolute value inequalities are and how they work. By following the steps outlined in this article, you should be able to solve similar inequalities on your own.

Frequently Asked Questions

What is an absolute value inequality? An absolute value inequality is an inequality that involves the absolute value of a quantity. It is a type of inequality that can be used to describe the distance of a quantity from zero on the number line.
How do I solve an absolute value inequality? To solve an absolute value inequality, you can split the inequality into two cases, one for the positive case and one for the negative case. You can then solve each case separately using basic algebraic manipulations.
What is the solution to the inequality $|x-4| \geq 9$ ? The solution to the inequality $|x-4| \geq 9$ is the interval $(-\infty, -5] \cup [13, \infty)$ .

Additional Resources

Absolute Value Inequalities: This article provides a comprehensive overview of absolute value inequalities, including how to solve them and how to express the solution as an interval.
Solving Absolute Value Inequalities: This article provides step-by-step instructions on how to solve absolute value inequalities, including how to split the inequality into two cases and how to combine the solutions.
Absolute Value Inequality Examples: This article provides examples of absolute value inequalities, including how to solve them and how to express the solution as an interval.

Introduction

In our previous article, we solved the inequality $|x-4| \geq 9$ and expressed the solution as an interval. In this article, we will answer some frequently asked questions about absolute value inequalities.

Q&A

Q: What is an absolute value inequality?

A: An absolute value inequality is an inequality that involves the absolute value of a quantity. It is a type of inequality that can be used to describe the distance of a quantity from zero on the number line.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you can split the inequality into two cases, one for the positive case and one for the negative case. You can then solve each case separately using basic algebraic manipulations.

Q: What is the solution to the inequality $|x-4| \geq 9$ ?

A: The solution to the inequality $|x-4| \geq 9$ is the interval $(-\infty, -5] \cup [13, \infty)$ .

Q: How do I know which case to use when solving an absolute value inequality?

A: When solving an absolute value inequality, you can use the following steps to determine which case to use:

Check the sign of the expression inside the absolute value: If the expression inside the absolute value is positive, use the positive case. If the expression inside the absolute value is negative, use the negative case.
Check the direction of the inequality: If the inequality is greater than or equal to, use the positive case. If the inequality is less than or equal to, use the negative case.

Q: Can I use absolute value inequalities to solve equations?

A: Yes, you can use absolute value inequalities to solve equations. However, you will need to use a different approach than solving absolute value inequalities. Instead of splitting the inequality into two cases, you can use the following steps to solve the equation:

Set the expression inside the absolute value equal to the constant: Set the expression inside the absolute value equal to the constant on the right-hand side of the equation.
Solve for the variable: Solve for the variable using basic algebraic manipulations.

Q: Can I use absolute value inequalities to solve systems of equations?

A: Yes, you can use absolute value inequalities to solve systems of equations. However, you will need to use a different approach than solving absolute value inequalities. Instead of splitting the inequality into two cases, you can use the following steps to solve the system of equations:

Solve one of the equations for one of the variables: Solve one of the equations for one of the variables using basic algebraic manipulations.
Substitute the expression into the other equation: Substitute the expression into the other equation.
Solve for the remaining variable: Solve for the remaining variable using basic algebraic manipulations.

Q: Are there any special cases to consider when solving absolute value inequalities?

A: Yes, there are several special cases to consider when solving absolute value inequalities. These include:

Zero on the right-hand side: If the right-hand side of the inequality is zero, the solution is the set of all real numbers.
Negative on the right-hand side: If the right-hand side of the inequality is negative, the solution is the empty set.
Equal to a constant: If the right-hand side of the inequality is equal to a constant, the solution is the set of all real numbers that are greater than or equal to the constant.

Conclusion

In this article, we have answered some frequently asked questions about absolute value inequalities. We have also provided some additional resources for further learning.

Additional Resources

Absolute Value Inequalities: This article provides a comprehensive overview of absolute value inequalities, including how to solve them and how to express the solution as an interval.
Solving Absolute Value Inequalities: This article provides step-by-step instructions on how to solve absolute value inequalities, including how to split the inequality into two cases and how to combine the solutions.
Absolute Value Inequality Examples: This article provides examples of absolute value inequalities, including how to solve them and how to express the solution as an interval.

Frequently Asked Questions

What is an absolute value inequality? An absolute value inequality is an inequality that involves the absolute value of a quantity. It is a type of inequality that can be used to describe the distance of a quantity from zero on the number line.
How do I solve an absolute value inequality? To solve an absolute value inequality, you can split the inequality into two cases, one for the positive case and one for the negative case. You can then solve each case separately using basic algebraic manipulations.
What is the solution to the inequality $|x-4| \geq 9$ ? The solution to the inequality $|x-4| \geq 9$ is the interval $(-\infty, -5] \cup [13, \infty)$ .

Glossary

Absolute value: The absolute value of a quantity is the distance of the quantity from zero on the number line.
Absolute value inequality: An inequality that involves the absolute value of a quantity.
Positive case: The case where the expression inside the absolute value is positive.
Negative case: The case where the expression inside the absolute value is negative.
Interval: A set of real numbers that includes all the numbers between two given numbers.