Writing Inequalities in Interval Notation

Oct 13, 2025 by recoveryshake.com 48 views

In mathematics, inequalities are used to describe relationships between numbers or expressions. Interval notation is a way to represent a set of numbers using a specific format. In this article, we will explore how to write an inequality in interval notation and graph the interval.

What is Interval Notation?

Interval notation is a way to represent a set of numbers using a specific format. It consists of a pair of numbers, one inside parentheses and the other inside brackets or parentheses. The numbers inside the parentheses represent the endpoints of the interval, while the numbers inside the brackets represent the inclusion or exclusion of the endpoints.

Writing Inequalities in Interval Notation

To write an inequality in interval notation, we need to follow these steps:

Determine the endpoints: Identify the numbers that are included in the inequality. If the inequality is greater than or equal to a number, the endpoint is included. If the inequality is less than or equal to a number, the endpoint is included.
Determine the type of bracket: If the endpoint is included, use a bracket. If the endpoint is not included, use a parenthesis.
Write the interval notation: Write the interval notation using the determined endpoints and brackets.

Example: Writing the Inequality in Interval Notation

Let's consider the inequality ${-4 \ \textless \ x \leq 5}$ . To write this inequality in interval notation, we need to follow the steps above.

Determine the endpoints: The endpoint -4 is not included, and the endpoint 5 is included.
Determine the type of bracket: Since the endpoint -4 is not included, we use a parenthesis. Since the endpoint 5 is included, we use a bracket.
Write the interval notation: The interval notation for the inequality ${-4 \ \textless \ x \leq 5}$ is ${(-4, 5]}$ .

Graphing the Interval

To graph the interval, we need to plot the endpoints on a number line and shade the region between them.

Plot the endpoints: Plot the points -4 and 5 on a number line.
Determine the direction of the shading: Since the inequality is less than, we shade the region to the left of the endpoint -4. Since the inequality is less than or equal to, we shade the region to the left of the endpoint -4 and include the point 5.
Graph the interval: The graph of the interval ${(-4, 5]}$ is a closed interval that includes the point 5 and excludes the point -4.

Conclusion

In this article, we explored how to write an inequality in interval notation and graph the interval. We learned how to determine the endpoints and the type of bracket to use in interval notation. We also learned how to graph the interval by plotting the endpoints and shading the region between them. By following these steps, we can write inequalities in interval notation and graph the intervals.

Common Inequalities and Their Interval Notation

Here are some common inequalities and their interval notation:

Inequality	Interval Notation
${x \ \textless \ 2}$	${(-\infty, 2)}$
${x \leq 3}$	${(-\infty, 3]}$
${x \ \textgreater \ 4}$	${(4, \infty)}$
${x \ \textgreater \ 5}$	${(5, \infty)}$

Practice Problems

Here are some practice problems to help you understand how to write inequalities in interval notation and graph the intervals:

Write the inequality ${x \ \textless \ 1}$ in interval notation.
Graph the interval ${(-2, 3)}$ .
Write the inequality ${x \leq 2}$ in interval notation.
Graph the interval ${(-\infty, 4)}$ .
Write the inequality ${x \ \textgreater \ 3}$ in interval notation.
Graph the interval ${(2, \infty)}$ .

Answer Key

Here are the answers to the practice problems:

The interval notation for the inequality ${x \ \textless \ 1}$ is ${(-\infty, 1)}$ .
The graph of the interval ${(-2, 3)}$ is a open interval that excludes the points -2 and 3.
The interval notation for the inequality ${x \leq 2}$ is ${(-\infty, 2]}$ .
The graph of the interval ${(-\infty, 4)}$ is a open interval that excludes the point 4.
The interval notation for the inequality ${x \ \textgreater \ 3}$ is ${(3, \infty)}$ .
The graph of the interval ${(2, \infty)}$ is a open interval that excludes the point 2.

Conclusion

In this article, we will answer some frequently asked questions about inequalities and interval notation.

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

A: An inequality is a statement that compares two expressions using a relation such as greater than, less than, greater than or equal to, or less than or equal to. An equation is a statement that says two expressions are equal.

Q: How do I determine the type of bracket to use in interval notation?

A: To determine the type of bracket to use in interval notation, you need to look at the inequality. If the inequality is greater than or equal to a number, use a bracket. If the inequality is less than or equal to a number, use a bracket. If the inequality is greater than a number, use a parenthesis. If the inequality is less than a number, use a parenthesis.

Q: What is the meaning of the symbol ${(-\infty, a)}$ ?

A: The symbol ${(-\infty, a)}$ represents an open interval that includes all real numbers less than a. In other words, it includes all numbers that are less than a, but does not include the number a itself.

Q: What is the meaning of the symbol ${(a, \infty)}$ ?

A: The symbol ${(a, \infty)}$ represents an open interval that includes all real numbers greater than a. In other words, it includes all numbers that are greater than a, but does not include the number a itself.

Q: How do I graph an interval?

A: To graph an interval, you need to plot the endpoints on a number line and shade the region between them. If the interval is open, do not include the endpoints in the shading. If the interval is closed, include the endpoints in the shading.

Q: What is the difference between a closed interval and an open interval?

A: A closed interval includes the endpoints, while an open interval does not include the endpoints.

Q: How do I write an inequality in interval notation?

A: To write an inequality in interval notation, you need to follow these steps:

Determine the endpoints of the inequality.
Determine the type of bracket to use for each endpoint.
Write the interval notation using the determined endpoints and brackets.

Q: What is the meaning of the symbol ${(-\infty, a)}$ in interval notation?

Q: What is the meaning of the symbol ${(a, \infty)}$ in interval notation?

Q: How do I determine the type of bracket to use in interval notation?

Q: What is the difference between a closed interval and an open interval?

A: A closed interval includes the endpoints, while an open interval does not include the endpoints.

Q: How do I graph an interval?

Conclusion

In this article, we answered some frequently asked questions about inequalities and interval notation. We covered topics such as the difference between an inequality and an equation, how to determine the type of bracket to use in interval notation, and how to graph an interval. By understanding these concepts, you can better write inequalities in interval notation and graph the intervals.

Common Mistakes to Avoid

Here are some common mistakes to avoid when working with inequalities and interval notation:

Not including the endpoints in the interval notation: Make sure to include the endpoints in the interval notation if they are included in the inequality.
Using the wrong type of bracket: Make sure to use the correct type of bracket for each endpoint.
Not shading the correct region: Make sure to shade the correct region on the number line when graphing an interval.
Not including the endpoints in the shading: Make sure to include the endpoints in the shading if the interval is closed.

Practice Problems

Here are some practice problems to help you understand how to write inequalities in interval notation and graph the intervals:

Write the inequality ${x \ \textless \ 2}$ in interval notation.
Graph the interval ${(-2, 3)}$ .
Write the inequality ${x \leq 2}$ in interval notation.
Graph the interval ${(-\infty, 4)}$ .
Write the inequality ${x \ \textgreater \ 3}$ in interval notation.
Graph the interval ${(2, \infty)}$ .

Answer Key

Here are the answers to the practice problems:

The interval notation for the inequality ${x \ \textless \ 2}$ is ${(-\infty, 2)}$ .
The graph of the interval ${(-2, 3)}$ is a open interval that excludes the points -2 and 3.
The interval notation for the inequality ${x \leq 2}$ is ${(-\infty, 2]}$ .
The graph of the interval ${(-\infty, 4)}$ is a open interval that excludes the point 4.
The interval notation for the inequality ${x \ \textgreater \ 3}$ is ${(3, \infty)}$ .
The graph of the interval ${(2, \infty)}$ is a open interval that excludes the point 2.