Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, write the interval in set-builder notation. \((4, \infty)\)

Step-by-Step Solution

Answer

The interval \((4, \infty)\) is expressed as \( \{ x \mid x > 4 \} \) in set-builder notation.

Step 1: Understand the Interval

The interval \((4, \infty)\) represents all numbers greater than 4 but does not include 4 itself. Infinity is a concept, not a number, so it is always open.

Step 2: Set-Builder Notation Format

Set-builder notation is written as \( \{ x \mid P(x) \} \), where \(P(x)\) is a condition that describes the elements of the set.

Step 3: Identify the Property Condition for the Interval

For the interval \((4, \infty)\), the condition \(P(x)\) is that \(x\) is greater than 4. This can be expressed as \(x > 4\).

Step 4: Write the Interval in Set-Builder Notation

Based on the condition identified, the interval in set-builder notation is \( \{ x \mid x > 4 \} \). This indicates the set of all real numbers \(x\) such that \(x\) is greater than 4.

Key Concepts

Interval Notation

Interval notation is a concise way to represent the set of numbers between two endpoints on the real number line.
It's especially useful because it succinctly shows whether the endpoints are included or not. In interval notation:

• Parentheses, \(( \text{ and } )\), are used for open intervals, indicating that the endpoints are not included.
• Brackets, \([ \text{ and } ]\), are used for closed intervals, indicating that the endpoints are included.
• A combination of parentheses and brackets can signify mixed intervals.

For example, the notation \((4, \infty)\) means all numbers greater than 4, but not including 4 itself, extending all the way to infinity.
Infinity, whether positive or negative, is always considered an open interval because it is a concept rather than a number.
Translating this interval to sentences, \((4, \infty)\) simply means "all numbers greater than 4." This interval is fundamental in understanding the way we discuss continuity and range in various mathematical contexts.

Inequalities

Inequalities are mathematical expressions used to demonstrate the relationship between two values, indicating that one value is larger or smaller than the other.
Symbols used in inequalities include:

• \(>\) for "greater than"
• \(<\) for "less than"
• \(\geq\) for "greater than or equal to"
• \(\leq\) for "less than or equal to"

In the example provided, \(x > 4\) is a simple inequality that signifies all real numbers greater than 4.
This is typically represented on the number line as an open circle at 4 with shading to the right, illustrating that 4 is not included in the set, but all numbers greater than 4 are.
Using inequalities, we can represent a wide range of conditions, whether solving equations or describing intervals, and they are crucial to forming set-builder notation as well as solving real-world problems.

Real Numbers

Real numbers encompass all the number types that can be plotted on a number line.
This includes:

• Positive numbers
• Negative numbers
• Zero
• Fractions
• Decimals
• Irrational numbers (like \(\pi\) or \(\sqrt{2}\))

Real numbers exclude imaginary numbers and representatively cover what we interact with in most real-world scenarios.
They are denoted by the symbol \( \mathbb{R} \), and any interval in math that includes real numbers can reflect any of these types.
For example, in set-builder notation such as \( \{ x \mid x > 4 \} \), the 'real numbers' specifier is often implied, meaning 'all real numbers \(x\) such that \(x > 4\)' are considered part of this set.
Whether working with discrete data or continuous ranges, real numbers form the backbone of most mathematical exercises and applications.