Consider The Following Interval: $[-4,6)$.1. State Whether The Interval Is Bounded Or Unbounded. - A. Bounded - B. Unbounded2. Represent The Interval With An Inequality. - A. $x \leq -4, X \ \textgreater \ 6$ - B.

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Introduction

In mathematics, intervals are used to represent a set of numbers within a specific range. These intervals can be either bounded or unbounded, depending on their endpoints. In this article, we will explore the concept of bounded and unbounded intervals, and how to represent them using inequalities.

What are Bounded and Unbounded Intervals?

A bounded interval is a set of numbers that has a finite range, meaning it has a clear beginning and end point. On the other hand, an unbounded interval is a set of numbers that has no finite range, meaning it extends infinitely in one or both directions.

Is the Interval $[-4,6)$ Bounded or Unbounded?

To determine whether the interval $[-4,6)$ is bounded or unbounded, we need to examine its endpoints. The interval starts at $-4$ and extends up to, but does not include, $6$. Since the interval has a clear beginning and end point, it is considered a bounded interval.

Answer: The interval $[-4,6)$ is a bounded interval.

Representing the Interval with an Inequality

To represent the interval $[-4,6)$ using an inequality, we need to consider the following:

  • The interval starts at $-4$, so we can write $x \geq -4$ to represent this endpoint.
  • The interval extends up to, but does not include, $6$, so we can write $x < 6$ to represent this endpoint.

Combining these two inequalities, we get:

−4≤x<6-4 \leq x < 6

This is the correct representation of the interval $[-4,6)$ using an inequality.

Answer: The correct representation of the interval $[-4,6)$ using an inequality is $-4 \leq x < 6$.

Conclusion

In conclusion, the interval $[-4,6)$ is a bounded interval, and it can be represented using the inequality $-4 \leq x < 6$. Understanding the concept of bounded and unbounded intervals is crucial in mathematics, as it helps us to analyze and solve problems involving intervals.

Key Takeaways

  • A bounded interval is a set of numbers that has a finite range.
  • An unbounded interval is a set of numbers that has no finite range.
  • The interval $[-4,6)$ is a bounded interval.
  • The correct representation of the interval $[-4,6)$ using an inequality is $-4 \leq x < 6$.

Further Reading

For more information on intervals and inequalities, we recommend checking out the following resources:

  • Khan Academy: Intervals and Inequalities
  • Math Is Fun: Intervals and Inequalities
  • Wolfram MathWorld: Intervals and Inequalities

Q: What is the difference between a bounded and an unbounded interval?

A: A bounded interval is a set of numbers that has a finite range, meaning it has a clear beginning and end point. On the other hand, an unbounded interval is a set of numbers that has no finite range, meaning it extends infinitely in one or both directions.

Q: How do I determine whether an interval is bounded or unbounded?

A: To determine whether an interval is bounded or unbounded, you need to examine its endpoints. If the interval has a clear beginning and end point, it is considered a bounded interval. If the interval extends infinitely in one or both directions, it is considered an unbounded interval.

Q: What is the correct representation of the interval $[-4,6)$ using an inequality?

A: The correct representation of the interval $[-4,6)$ using an inequality is $-4 \leq x < 6$.

Q: Can an interval be both bounded and unbounded at the same time?

A: No, an interval cannot be both bounded and unbounded at the same time. An interval is either bounded or unbounded, but not both.

Q: How do I represent an unbounded interval using an inequality?

A: To represent an unbounded interval using an inequality, you need to use a combination of symbols to indicate that the interval extends infinitely in one or both directions. For example, the interval $(-\infty, 6)$ can be represented using the inequality $x < 6$.

Q: Can I have a negative and a positive infinity in an interval?

A: No, you cannot have a negative and a positive infinity in an interval. Infinity is a concept that represents a value that is larger than any other value, but it is not a specific number. Therefore, you cannot have both a negative and a positive infinity in an interval.

Q: How do I represent an interval with a negative infinity?

A: To represent an interval with a negative infinity, you can use the symbol $(-\infty, a)$, where $a$ is a specific number. For example, the interval $(-\infty, 6)$ can be represented using the inequality $x < 6$.

Q: How do I represent an interval with a positive infinity?

A: To represent an interval with a positive infinity, you can use the symbol $(a, \infty)$, where $a$ is a specific number. For example, the interval $(6, \infty)$ can be represented using the inequality $x > 6$.

Q: Can I have an interval with a negative infinity and a positive infinity?

A: Yes, you can have an interval with a negative infinity and a positive infinity. For example, the interval $(-\infty, \infty)$ represents all real numbers.

Q: How do I represent an interval with a negative infinity and a positive infinity?

A: To represent an interval with a negative infinity and a positive infinity, you can use the symbol $(-\infty, \infty)$. For example, the interval $(-\infty, \infty)$ represents all real numbers.

Conclusion

In conclusion, understanding the concept of bounded and unbounded intervals is crucial in mathematics. By knowing how to represent intervals using inequalities, you will be better equipped to solve problems involving intervals and inequalities.