Problem 9 - College Algebra with Modeling and Visualization

Question

Express each of the following in interval notation. $$ \\{x | x<1 \text { or } x \geq 3\\} $$

Step-by-Step Solution

Answer

\((-\infty, 1) \cup [3, \infty)\).

Step 1: Understanding the Set Notation

The set notation \( \{x | x<1 \text{ or } x \geq 3\} \) describes a set of numbers. It includes numbers less than 1 and numbers greater than or equal to 3.

Step 2: Identifying the Intervals

We note that there are two separate conditions involved: \( x < 1 \) and \( x \geq 3 \). These give us two intervals. The first interval is all numbers less than 1, and the second interval is all numbers that are 3 or greater.

Step 3: Writing Each Interval

The interval for \( x < 1 \) is \((-\infty, 1)\) because it includes all numbers less than 1. The interval for \( x \geq 3 \) is \([3, \infty)\) because it includes all numbers beginning from 3 going onwards.

Step 4: Combining Intervals with Union

Since the expression uses 'or', we join these intervals using a union operation. The intervals \( (-\infty, 1) \) and \( [3, \infty) \) combined are written as \((-\infty, 1) \cup [3, \infty)\).

Key Concepts

Set Notation

Set notation is a mathematical way of denoting a set of numbers or objects that meet a specific condition. In the exercise provided, the set notation \( \{x | x<1 \text{ or } x \geq 3\} \) is used. This notation means: "the set of all \( x \) such that \( x \) is less than 1 or \( x \) is greater than or equal to 3." The vertical line \( | \) means "such that," and it introduces the condition that defines the set.
The set notation effectively describes all the members included in the set using inequalities (in this case) to set boundaries.

• \( x<1 \) includes all numbers less than 1.
• \( x \geq 3 \) covers all numbers starting from 3 and extending upwards.

This approach allows a clear and precise way to express collections of numbers, often used in conjunction with other mathematical concepts like intervals.

Union of Intervals

The concept of a union of intervals is essential when dealing with sets that involve more than one condition. When you have multiple sets described with conditions like "or," you use the union to bring these sets together. In interval notation, the union operation is denoted by \( \cup \).
In the problem statement, two separate intervals are derived from the conditions:

• \( (-\infty, 1) \), for all numbers less than 1
• \( [3, \infty) \), for all numbers 3 and greater

To combine these intervals, we use the union:

• The complete set becomes \( (-\infty, 1) \cup [3, \infty) \).

Union essentially merges multiple sets into one, encompassing all elements that belong to any of the original sets. This operation is particularly useful in expressing solutions to inequalities where multiple ranges need to be considered simultaneously.

Inequalities

Inequalities are mathematical expressions used to compare two values or expressions, often indicating that one is larger or smaller than another. They are foundational to expressing conditions within set and interval notations.
In the provided exercise, two inequalities are mentioned:

• \( x < 1 \): This implies that \( x \) can be any number less than 1.
• \( x \geq 3 \): This means \( x \) includes 3 and any number larger.

Inequalities are represented using symbols like \(<\), \(>\), \(\leq\), and \(\geq\). When solving exercises involving inequalities:

• Identify each condition separately.
• Convert each into intervals as needed.
• Consider using union \( \cup \) for combinations of separate inequalities.

In mathematical contexts, inequalities help define boundaries and ranges, making them a critical part of understanding set notation and interval combinations effectively.