Problem 16 - Precalculus

Question

Write the set using interval notation. $$ \\{x \mid x \leq-3 \text { or } x>0\\} $$

Step-by-Step Solution

Answer

The interval notation is \( ( -\infty, -3 ] \cup ( 0, \infty ) \).

Step 1: Identify the Components

First, let's identify what the set \( \{x \mid x \leq -3 \text { or } x > 0\} \) means. This is a set of all \( x \) such that \( x \) is less than or equal to -3, or \( x \) is greater than 0.

Step 2: Express as Two Intervals

We have two conditions here:1. \( x \leq -3 \) corresponds to the interval \(( -\infty, -3 ]\).2. \( x > 0 \) corresponds to the interval \(( 0, \infty )\).

Step 3: Combine the Intervals

Since the original set uses the word 'or', we need a union of the two intervals. In interval notation, this is expressed as \[ ( -\infty, -3 ] \cup ( 0, \infty ) \].

Step 4: Final Result

The set in interval notation is \[ ( -\infty, -3 ] \cup ( 0, \infty ) \]. This captures all numbers less than or equal to -3 or greater than 0.

Key Concepts

Set Notation

When dealing with expressions like \( \{x \mid x \leq -3 \text{ or } x > 0\} \), it is crucial to understand the concept of set notation. Set notation is a way to describe a collection of numbers that share certain properties. In this case, it describes all numbers \( x \) that meet at least one of the given inequalities. Set notation provides a concise way to express complex logical statements about numbers and their relationships.

In logic and mathematics, the symbol \( \mid \) stands for "such that" and separates the variable from the condition. It helps to define the elements of the set precisely. Set notation is valuable as it can depict both explicit and implicit characteristics of a group of numbers. By using set notation, mathematicians and students alike can handle a vast array of mathematical statements, equations, and functions in an organized fashion.

Inequalities

Inequalities are mathematical expressions that indicate the relative size or order of two values. They use symbols such as \( < \), \( \leq \), \( > \), or \( \geq \) to express these relationships. In the given exercise, inequalities specify conditions under which certain numbers belong to the set. For example:

• \( x \leq -3 \) means any number \( x \) that is less than or equal to -3 is part of the set.
• \( x > 0 \) indicates that any number \( x \) greater than 0 qualifies for inclusion in the set.

Inequalities are useful to construct intervals, which can then be expressed using interval notation. They allow us to define precise ranges of numbers that share specific numerical properties, thereby acting as a bridge between simple arithmetic comparisons and more complex algebraic expressions.

Union of Intervals

Combining sets of numbers using the concept of union is a vital component of set theory and interval notation. The union, denoted by the symbol \( \cup \), signifies an operation where all elements of combined intervals are included in the result. In our case, it's crucial to merge two separate intervals into a single statement.

To execute the union of intervals, we take the results from each defined condition:\[\begin{align*}\text{Interval for } x \leq -3 &: \ ( -\infty, -3 ] \text{Interval for } x > 0 &: \ ( 0, \infty )\end{align*}\]When united, these intervals become \( ( -\infty, -3 ] \cup ( 0, \infty ) \), which represents all numbers that are either less than or equal to -3 or greater than 0. This union process ensures that the combined interval covers all possible solutions to the original conditions. The use of \'union\' here highlights the inclusive approach of set operations to encapsulate diverse element satisfaction within a single expression.