Problem 43 - College Algebra

Question

For the following exercises, write the set in interval notation. $$ \\{x |-1< x <3\\} $$

Step-by-Step Solution

Answer

\((-1, 3)\)

Step 1: Understand Set Notation

The given set \( \{x \,|-1 < x < 3\} \) describes a range of real numbers where \( x \) must be greater than \(-1\) and less than \(3\). This is a description in set-builder notation.

Step 2: Translate to Interval Notation

Interval notation expresses the same set of numbers as \((-1, 3)\). The parentheses \((\) and \()\) indicate that the endpoints \(-1\) and \(3\) are not included in the set.

Key Concepts

Set-Builder Notation

Set-builder notation is a concise way to describe a set of numbers that satisfy a particular condition. It is written in the form \( \{ x \,|\, ~\text{condition on} ~ x \} \). This notation clearly outlines the rules that numbers must meet to be included in the set. In the given example, \( \{ x \,|\, -1 < x < 3 \} \), we are describing all real numbers \( x \) that are greater than \(-1\) but less than \(3\).
This notation is handy for solving problems because it lays out the criteria a variable must fit within a defined range. Here's how to read it better:

• The vertical line \( | \) represents "such that," which means it lays down the condition for the element \( x \).
• The part \(-1 < x < 3\) provides the specific range of \( x \).

Understanding set-builder notation is crucial because it provides a clear and mathematical way to delineate sets.

Real Numbers

Real numbers are a crucial concept in mathematics that you encounter often. They include all the numbers on the number line, encompassing both rational and irrational numbers. Rational numbers can be expressed as fractions (like \( \frac{1}{2} \) or \( 0.75 \)), while irrational numbers cannot, such as \( \pi \) or \( \sqrt{2} \).
When working with intervals, like in the exercise, the value \( x \) belongs to the set of real numbers. This means that any number in the range described by the interval can be a potential solution. Real numbers can be:

• Positive, negative, or zero
• Whole numbers, fractions, or decimals
• Finite or infinite (meaning they can stretch indefinitely in positive or negative directions)

Thinking in terms of real numbers allows you to cover every possibility along the continuum of the number line, which is essential in mathematics problems involving intervals.

Endpoints

Endpoints are significant when discussing intervals, especially distinguishing between open and closed intervals. In the context of interval notation, endpoints mark the boundaries of the number range.
For example, in the interval \((-1, 3)\), the numbers \(-1\) and \(3\) are the endpoints. However, since the interval uses parentheses, these numbers are not included in the set. Parentheses \((\),\()\) denote an open interval, indicating exclusion of the endpoints from the set.
In contrast, brackets \([\),\()]\) denote a closed interval, meaning the endpoints are part of the set.
Key points about endpoints include:

• Determining whether an endpoint is included or excluded depends on the type of bracket used.
• Understanding endpoints helps accurately interpret and translate between different types of mathematical notations.
• Endpoints provide precise limitations, crucial for clearly defining a solution space.

Knowing how to handle endpoints is essential for precision in problems involving ranges and intervals.