Problem 48 - College Algebra

Question

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

Step-by-Step Solution

Answer

\( \{ x \ | \ x > 4 \} \)

Step 1: Identify the Interval

The interval given is \((4, \infty)\), which represents all the numbers greater than 4 but not including 4 itself, and extending indefinitely to positive infinity.

Step 2: Express in Set-Builder Notation

To write the interval \((4, \infty)\) in set-builder notation, we need to define the set using a variable and a condition. In this case, we will use the variable \(x\) and the condition will be \(x > 4\). Thus, the set-builder notation is \( \{ x \ | \ x > 4 \} \).

Key Concepts

Interval Notation

Interval notation is a way of describing a set of numbers along a number line. It is a concise and clear method for denoting a certain range of numbers such as those that fall between two endpoints. In this context, intervals are used to describe either a continuous range of numbers or all numbers from a certain point onward. There are two main types of intervals: open and closed. An open interval,

• for example, \((a, b)\), indicates that the endpoints \(a\) and \(b\) are not included in the set.
• A closed interval, such as \([a, b]\), means both endpoints are included.

In the given exercise, the interval

• \((4, \infty)\) tells us all numbers greater than 4 but not including 4 are part of this set.
• Infinity, in this case, means there is no upper bound.

We use parentheses to represent open intervals because they visually remind us that the endpoint is not part of the set. This way, interval notation provides a quick and useful shorthand for describing unbounded and bounded sets.

Inequalities

Inequalities are mathematical statements that involve the symbols like ">", "<", ">=", and "<=". They tell us how different numbers or expressions compare in size. In other words, they define relationships where quantities are not equal.

• An inequality such as \(x > 4\) tells us that \(x\) is greater than 4.

Inequalities can be both strict (using only ">" or "<") or non-strict (using ">=" or "<="). When we express intervals in set-builder notation, we are essentially translating the interval into a statement of inequality.For example:

• The interval \((4, \infty)\) translates to the inequality \(x > 4\).

The inequality expresses that \(x\) can be any number larger than 4, emphasizing the open nature of the point at 4 in the original interval. This connection between intervals and inequalities is fundamental to understanding how numbers relate within a specified range. It helps us make sense of mathematical statements and solve problems involving ranges and boundaries.

Infinity in Mathematics

Infinity is a fascinating concept in mathematics that serves as a symbol for numbers that have no limit. It is not a number in the conventional sense, but rather a way of describing something without an endpoint or limit.In terms of notation and use:

• Infinity is represented by the symbol \(\infty\).
• It indicates that a set of numbers extends endlessly in one direction, either positive or negative.

In intervals:

• The expression \((4, \infty)\) uses infinity to show that there is no upper limit to the numbers greater than 4.

Infinity is used in calculus, set theory, and other mathematical fields to represent concepts of limitless quantity and continuous growth. It's crucial to grasp that while infinity can describe the size or extent of a set, no actual number can reach infinity. In expressions, infinity provides a way to communicate endlessly large or small quantities and to understand limits better in mathematics.