Write The Following Inequalities In Interval Notation. Type infinity For $\infty$.a) $x \leq 0$b) $x \ \textless \ 3.2$c) $x \geq \frac{5}{2}$

Introduction

Interval notation is a way of writing inequalities in a concise and visual format. It is commonly used in mathematics to represent the solution sets of inequalities. In this article, we will learn how to write the given inequalities in interval notation.

Inequality a) $x \leq 0$

The inequality $x \leq 0$ represents all real numbers that are less than or equal to zero. In interval notation, this can be written as:

$(-\infty, 0]$

This notation indicates that the solution set includes all real numbers less than zero, including zero itself.

Inequality b) $x \ \textless \ 3.2$

The inequality $x \ \textless \ 3.2$ represents all real numbers that are less than 3.2. In interval notation, this can be written as:

$(-\infty, 3.2)$

This notation indicates that the solution set includes all real numbers less than 3.2, but does not include 3.2 itself.

Inequality c) $x \geq \frac{5}{2}$

The inequality $x \geq \frac{5}{2}$ represents all real numbers that are greater than or equal to $\frac{5}{2}$. In interval notation, this can be written as:

$[\frac{5}{2}, \infty)$

This notation indicates that the solution set includes all real numbers greater than or equal to $\frac{5}{2}$, including $\frac{5}{2}$ itself.

Understanding Interval Notation

Interval notation is a way of representing the solution sets of inequalities in a concise and visual format. It uses square brackets and parentheses to indicate the inclusion or exclusion of endpoints.

• A square bracket $[$ indicates that the endpoint is included in the solution set.
• A parenthesis $($ indicates that the endpoint is not included in the solution set.
• A negative infinity $-\infty$ indicates that the solution set includes all real numbers less than the endpoint.
• A positive infinity $\infty$ indicates that the solution set includes all real numbers greater than the endpoint.

Examples of Interval Notation

Here are some examples of interval notation:

• $(-\infty, 2)$ represents all real numbers less than 2.
• $(2, \infty)$ represents all real numbers greater than 2.
• $[2, 5]$ represents all real numbers greater than or equal to 2 and less than or equal to 5.
• $(-\infty, 2] \cup [5, \infty)$ represents all real numbers less than 2 and all real numbers greater than or equal to 5.

Conclusion

Interval notation is a powerful tool for representing the solution sets of inequalities. By using square brackets and parentheses, we can concisely and visually represent the inclusion or exclusion of endpoints. In this article, we learned how to write the given inequalities in interval notation and gained a deeper understanding of interval notation.

Key Takeaways

• Interval notation is a way of representing the solution sets of inequalities in a concise and visual format.
• A square bracket $[$ indicates that the endpoint is included in the solution set.
• A parenthesis $($ indicates that the endpoint is not included in the solution set.
• A negative infinity $-\infty$ indicates that the solution set includes all real numbers less than the endpoint.
• A positive infinity $\infty$ indicates that the solution set includes all real numbers greater than the endpoint.

Practice Problems

Try to write the following inequalities in interval notation:

• $x \geq -1$
• $x \ \textless \ 4.5$
• $x \leq \frac{3}{4}$

Answer Key

• $[-1, \infty)$
• $(-\infty, 4.5)$
• $(-\infty, \frac{3}{4}]$
  Interval Notation Q&A =========================

Frequently Asked Questions

Q: What is interval notation?

A: Interval notation is a way of representing the solution sets of inequalities in a concise and visual format. It uses square brackets and parentheses to indicate the inclusion or exclusion of endpoints.

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

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

1. Determine the direction of the inequality (less than, less than or equal to, greater than, or greater than or equal to).
2. Identify the endpoint of the inequality (a specific value or infinity).
3. Use square brackets $[$ to indicate that the endpoint is included in the solution set.
4. Use parentheses $($ to indicate that the endpoint is not included in the solution set.
5. Use negative infinity $-\infty$ to indicate that the solution set includes all real numbers less than the endpoint.
6. Use positive infinity $\infty$ to indicate that the solution set includes all real numbers greater than the endpoint.

Q: What is the difference between a square bracket and a parenthesis in interval notation?

A: A square bracket $[$ indicates that the endpoint is included in the solution set, while a parenthesis $($ indicates that the endpoint is not included in the solution set.

Q: How do I represent negative infinity in interval notation?

A: To represent negative infinity in interval notation, use the symbol $-\infty$.

Q: How do I represent positive infinity in interval notation?

A: To represent positive infinity in interval notation, use the symbol $\infty$.

Q: Can I have multiple intervals in a single expression?

A: Yes, you can have multiple intervals in a single expression. To do this, use the union symbol $\cup$ to combine the intervals.

Q: How do I write an inequality with multiple intervals?

A: To write an inequality with multiple intervals, follow these steps:

1. Write each interval separately using interval notation.
2. Use the union symbol $\cup$ to combine the intervals.

Q: What is the union of two intervals?

A: The union of two intervals is the set of all real numbers that are in either of the two intervals.

Q: How do I find the union of two intervals?

A: To find the union of two intervals, follow these steps:

1. Identify the endpoints of each interval.
2. Determine which endpoints are included in the solution set.
3. Use square brackets $[$ to indicate that the endpoints are included in the solution set.
4. Use parentheses $($ to indicate that the endpoints are not included in the solution set.
5. Use negative infinity $-\infty$ to indicate that the solution set includes all real numbers less than the endpoint.
6. Use positive infinity $\infty$ to indicate that the solution set includes all real numbers greater than the endpoint.

Q: Can I have empty intervals in interval notation?

A: Yes, you can have empty intervals in interval notation. An empty interval is represented by a single point, such as $\{2\}$.

Q: How do I represent an empty interval in interval notation?

A: To represent an empty interval in interval notation, use a single point, such as $\{2\}$.

Conclusion

Interval notation is a powerful tool for representing the solution sets of inequalities. By understanding the basics of interval notation, you can concisely and visually represent the inclusion or exclusion of endpoints. In this article, we answered frequently asked questions about interval notation and provided examples to help you understand the concepts.

Key Takeaways

• Interval notation is a way of representing the solution sets of inequalities in a concise and visual format.
• A square bracket $[$ indicates that the endpoint is included in the solution set.
• A parenthesis $($ indicates that the endpoint is not included in the solution set.
• A negative infinity $-\infty$ indicates that the solution set includes all real numbers less than the endpoint.
• A positive infinity $\infty$ indicates that the solution set includes all real numbers greater than the endpoint.
• The union of two intervals is the set of all real numbers that are in either of the two intervals.

Practice Problems

Try to answer the following questions:

• What is the difference between a square bracket and a parenthesis in interval notation?
• How do I represent negative infinity in interval notation?
• How do I represent positive infinity in interval notation?
• Can I have multiple intervals in a single expression?
• How do I write an inequality with multiple intervals?

Answer Key

• A square bracket $[$ indicates that the endpoint is included in the solution set, while a parenthesis $($ indicates that the endpoint is not included in the solution set.
• To represent negative infinity in interval notation, use the symbol $-\infty$.
• To represent positive infinity in interval notation, use the symbol $\infty$.
• Yes, you can have multiple intervals in a single expression.
• To write an inequality with multiple intervals, follow these steps: 1. Write each interval separately using interval notation. 2. Use the union symbol $\cup$ to combine the intervals.