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Chapter 2: Problem 51

Express each set in the simplest interval form. $$ (-\infty, 3) \cup(-\infty,-2) $$

Short Answer

Expert verified
The simplest interval is \((-\infty, 3)\).

Step by step solution

01

Understand the Union of Intervals

The given expression represents the union of two intervals: \((-\infty, 3)\) and \((-\infty, -2)\). The union of these intervals includes all numbers that are in either \((-\infty, 3)\) or \((-\infty, -2)\).
02

Combine Intervals

Since \((-\infty, 3)\) and \((-\infty, -2)\) are both intervals that extend to negative infinity, their union will also extend to negative infinity. This means that any number left of 3 or 3 itself falls within the overall interval.
03

Simplify Interval

Combine the intervals into a single interval. Since \((-\infty, 3)\) is a larger interval that encompasses all the values in \((-\infty, -2)\), the simplest form of the interval is \((-\infty, 3)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

interval notation
Interval notation is a way of representing a range of numbers on a number line. It is very useful in mathematics. In interval notation, we use brackets and parentheses to indicate the start and end points of an interval.

Here's a quick overview:
- **Parentheses, like \(a, b\),** mean that the interval does not include the endpoints **a** and **b**.
- **Brackets, like \[a, b\],** mean that the interval includes the endpoints **a** and **b**.

For example:
- \(1, 4\) includes all the numbers between 1 and 4 but not including 1 and 4.
- \[-2, 3\] includes all the numbers between -2 and 3, including -2 and 3 themselves.
union of intervals
The union of intervals is a way to combine multiple intervals into one. When we talk about the union of two intervals, we mean all numbers that are in either one interval or the other.

In mathematical terms, if we have two intervals A and B, the union of A and B, written as \(A \cup B\), includes any number that is in A, B, or both.

In our exercise:
- We have \((-\infty, 3) \cup(-\infty, -2)\).
- We need to think about all the numbers that are either in \((-\infty, 3)\) or \((-\infty, -2)\) or both.

Since both intervals go to negative infinity, and the overall interval cannot go beyond 3, the union is simplified.
simplifying intervals
Simplifying intervals means combining multiple intervals into the simplest form.

In the exercise we have two intervals that extend to negative infinity:
\((-\infty, 3) \) and \((-\infty, -2)\).
To simplify them, we can look at the larger interval that encompasses the smaller one.

Since \((-\infty, 3)\) includes all the numbers in \((-\infty, -2)\), the simplest form of the interval becomes \((-\infty, 3)\).

By combining these steps, we see that simplifying intervals can often mean recognizing one interval includes the other and using the larger one to represent the entire range.

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