Problem 10 - College Algebra with Modeling and Visualization

Question

Express each of the following in interval notation. $$ \\{x | x \leq-2 \text { or } x \geq 0\\} $$

Step-by-Step Solution

Answer

The interval notation is \((-\infty, -2] \cup [0, \infty)\).

Step 1: Understanding Set Notation

The given set \( \{x | x \leq -2 \text{ or } x \geq 0 \} \) represents all \( x \) such that \( x \) is less than or equal to -2 or greater than or equal to 0. This describes two separate ranges of numbers on the number line.

Step 2: Define the Intervals

We need to determine two separate intervals from the given set notation: The first interval includes all numbers less than or equal to -2, which is written as \((-\infty, -2]\). The second interval includes all numbers greater than or equal to 0, which is written as \([0, \infty)\).

Step 3: Combine Intervals with Union

Since the problem specifies \( x \leq -2 \text{ or } x \geq 0 \), these intervals should be combined using the union sign \( \cup \). Therefore, the complete interval in interval notation is \((-\infty, -2] \cup [0, \infty)\).

Key Concepts

Set Notation

Set notation is a way to describe a group or collection of numbers based on specific conditions or rules. It's like creating a rule book that tells us which numbers belong in our set. In the given exercise, set notation is written as \( \{ x | x \leq -2 \text{ or } x \geq 0 \} \). This notation consists of:

• An opening curly brace \( \{ \), which indicates the beginning of our set.
• A vertical bar \( | \), which means "such that". This tells us the conditions that elements of the set must satisfy.
• The condition \( x \leq -2 \text{ or } x \geq 0 \), which describes the values \( x \) can take.

This expression represents all real numbers \( x \) that are either less than or equal to -2, or greater than or equal to 0. There is a logical "or" in between two separate conditions, indicating that if a number satisfies either of the conditions, it belongs in the set.

Union of Intervals

When dealing with intervals on the number line, we often have to combine or "unite" different segments to represent a set of numbers. This is called the union of intervals. It's symbolized by \( \cup \), which signifies the combination of two or more intervals into one.In our exercise, we have two distinct intervals:

• The first interval, \((\! -\infty, -2] \), includes all numbers that are less than or equal to \(-2\).
• The second interval, \([0, \infty) \), includes all numbers greater than or equal to \(0\).

The union of these intervals is expressed as \((\! -\infty, -2] \cup [0, \infty)\). This means any number within these combined intervals is part of the set. The use of the union ensures that both conditions \( x \leq -2 \) and \( x \geq 0 \) are accounted for in the final expression.

Number Line Representation

Visualizing intervals on a number line can help us better understand how these intervals relate to each other. A number line is a straight horizontal line with numbers placed at intervals, providing a visual representation of numerical values and their relationships.To represent the intervals from our exercise on a number line, follow these steps:

• Draw a horizontal line and mark critical points, such as \(-2\) and \(0\).
• For the interval \((\! -\infty, -2]\), draw a solid dot at \(-2\) and shade the line extending leftwards, indicating all numbers are included up to and including \(-2\).
• For the interval \([0, \infty)\), place a solid dot at \(0\) and shade the line extending rightwards, indicating all numbers from \(0\) and beyond are included.

Using a number line representation helps visualize that no numbers between \(-2\) and \(0\) are included in our set, highlighting the separation between the two intervals described by the union.