Problem 43 - College Algebra

Question

Write the set in interval notation. $$ \\{x \mid-1

Step-by-Step Solution

Answer

\((-1, 3)\)

Step 1: Understand the Set Notation

The given set \( \{ x \mid -1 < x < 3 \} \) is presented in set-builder notation. It describes all numbers \( x \) such that \( x \) is greater than \(-1\) and less than \(3\).

Step 2: Identify the Interval Type

Since the given condition involves strict inequalities \(-1 < x < 3\), we are dealing with an open interval. Open intervals do not include their endpoints.

Step 3: Write in Interval Notation

Since the set is described by \(-1 < x < 3\), we translate this into interval notation as \((-1, 3)\). The parentheses indicate that the endpoints \(-1\) and \(3\) are not included in the interval.

Key Concepts

set-builder notation

Set-builder notation is a way of describing a set by specifying the properties that its members must satisfy. It looks like this: \( \{ x \mid \text{condition} \} \). This is read as "the set of all \( x \) such that the condition is true."

In the example \( \{ x \mid -1 < x < 3 \} \), the set includes all numbers \( x \) that are greater than \(-1\) but less than \(3\). Here, the condition is a range defined by strict inequalities.

Set-builder notation is particularly useful when describing sets with many elements or elements that follow a specific logical rule. It allows you to clearly define which numbers are included in the set without having to list them all. As this method often includes symbols and math terms, understanding the fundamental logical connectors like "such that" is vital for interpreting a set correctly.

open interval

An open interval is a type of interval that does not include its endpoints. You will often encounter open intervals in mathematics, especially when dealing with range specifications.

Open intervals are denoted with parentheses. For example, in interval notation, the open interval \((-1, 3)\) means that this interval includes all numbers strictly between \(-1\) and \(3\).

• "(" and ")" imply the values at the endpoints themselves are not part of the set.
• This is different from closed intervals, which are denoted by square brackets "[" and "]" and do include their endpoints.

Because open intervals exclude endpoints, they are useful in calculus and mathematical analysis, where continuity and limits are considered. Understanding their properties simplifies many aspects of higher-level mathematics.

strict inequalities

Strict inequalities describe situations where one number is either greater than or less than another but not equal to it. These are represented using symbols like "<" for "less than" and ">" for "greater than".

In \(-1 < x < 3\), both parts of the inequality are strict. This means \( x \) can never be exactly \(-1\) or \(3\), but must be somewhere in between these two numbers.

• Strict inequalities are visually represented with open circles on a number line, indicating that the endpoints aren't included.
• They define the kind of interval (in this case, open) you'll be working with.

Grasping strict inequalities is an essential skill in algebra and other mathematical disciplines, as it helps interpret valid ranges of solutions to equations and inequalities. They are crucial when applying mathematical functions, limits, and even during optimization problems.