Problem 45 - College Algebra

Question

Write the set in interval notation. $$ \\{x \mid x<4\\} $$

Step-by-Step Solution

Answer

The interval notation is \((-\infty, 4)\).

Step 1: Identify the Type of Inequality

The given set is described by the inequality \( x < 4 \). This is a simple inequality indicating that \( x \) can be any number less than 4.

Step 2: Determine the Open or Closed Boundaries

Since the inequality is \( x < 4 \), the number 4 is not included in the set. This means we will use an open bracket (parenthesis) at 4, noted as \( ) \).

Step 3: Identify the Extent of the Solution Set

The inequality \( x < 4 \) means x can be any number from \(-\infty\) to 4, but does not include 4. Because \(-\infty\) is not a number, it will always be paired with a parenthesis.

Step 4: Write in Interval Notation

Using the information from the previous steps, we write the interval notation as \((-\infty, 4)\). This notation shows that the set includes all numbers less than 4, starting from negative infinity and not including 4 itself.

Key Concepts

Inequalities

Inequalities help us understand the relationships between values. In the expression \( x < 4 \), the symbol \(<\) indicates that \( x \) is any number that is less than 4. Such expressions can be useful in defining sets of numbers, representing them in various contexts, like graphs or number lines.
Understanding the inequality symbols is key:

• \(<\) means 'less than'.
• \(>\) means 'greater than'.
• \(\leq\) means 'less than or equal to'.
• \(\geq\) means 'greater than or equal to'.

Each symbol tells us how to compare values and which numbers fall within a certain range. In our example, all numbers less than 4 satisfy the inequality.

Open and Closed Boundaries

When expressing ranges of numbers, knowing whether the boundary is open or closed is essential. An 'open boundary' means the endpoint is not part of the set, whereas a 'closed boundary' means it is included.
In interval notation:

• Parentheses \(( )\) denote open boundaries. The number next to the parenthesis is not included in the interval.
• Brackets \([ ]\) denote closed boundaries, indicating the number is part of the interval.

For the inequality \( x < 4 \), because 4 is not included, it uses the open boundary notation with a parenthesis: \((-\infty, 4)\). Remember, open boundaries are always used when the interval stretches to infinity.

Infinity in Mathematics

Infinity is a concept used to describe something that is without bound or limit. In mathematics, infinity isn't a number but is used to capture the idea of endlessness.
When writing intervals:

• Using \( \infty \) or \(-\infty \) indicates that the range continues indefinitely in a positive or negative direction, respectively.
• Infinity is always paired with a parenthesis, representing an open boundary because it isn't a specific, reachable value.

In our notation \(( -\infty, 4)\), it tells us the number set starts from negative infinity and goes up to, but doesn't include, 4. Understanding infinity helps describe unlimited or unbounded sets in mathematics effectively.