Problem 1 - College Algebra with Modeling and Visualization

Question

Express the following in interval notation. $$ x<2 $$

Step-by-Step Solution

Answer

The interval notation for \( x < 2 \) is \((-\infty, 2)\).

Step 1: Identify the Inequality Type

We are given the inequality \( x < 2 \). This means that \( x \) can take any value less than 2. We need to express this in interval notation.

Step 2: Determine Interval Bounds

Since \( x \) is less than 2, it does not include 2 itself. Therefore, the upper bound of our interval should be 2, but not inclusive. The lower bound of the interval is negative infinity, as the variable is allowed to take any smaller value.

Step 3: Express in Interval Notation

Interval notation uses parentheses or brackets to denote open or closed intervals. Here, since 2 is not inclusive, we use a parenthesis. Negative infinity is also not a real number, hence always uses a parenthesis. Therefore, the interval notation for \( x < 2 \) is \((-\infty, 2)\).

Key Concepts

Inequality

An inequality is a mathematical statement that compares two expressions and shows whether one quantity is greater than, less than, or equal to another. Using inequality symbols, we can represent relationships like **less than** (<), **greater than** (>), **less than or equal to** (≤), and **greater than or equal to** (≥). Each of these symbols helps to express how one value relates to another, which is crucial for solving equations and understanding math concepts.

For example, in the inequality \( x < 2 \), the symbol \(<\) indicates that \( x \) can take any value less than 2. Inequalities allow flexibility in which numbers can satisfy the condition, offering a range of possible solutions. This characteristic of inequalities is what distinguishes them from equalities, where only one specific number satisfies the condition.

Inequality Type

Understanding the type of inequality is key to solving mathematical problems. Different types of inequalities include:

• **Strict Inequalities**: These use the symbols < or >, meaning the values on either side are not equal (e.g., \( x < 2 \)).
• **Non-strict Inequalities**: These include ≤ or ≥, allowing for equality, so the values on either side can be equal (e.g., \( x ≤ 2 \)).

For our exercise, we are working with a strict inequality, \( x < 2 \). This means that \( x \) can be any value less than, but not equal to, 2. Recognizing the type of inequality helps determine how we express it in interval notation and understand the limitations they pose on the variable being studied.

Interval Bounds

Interval bounds are the endpoints of the range of values that a variable can take. In inequalities, they help define the limits of values.

For the inequality \( x < 2 \), the interval bounds are:

• **Lower Bound**: Negative infinity (\(-\infty\)), since there is no smallest value that \( x \) can not be. The variable can take increasingly smaller values without bound.
• **Upper Bound**: 2, since \( x \) can take any value less than 2, but not equal to 2.

Interval bounds are critical because they allow us to graph or describe the solution set effectively. In interval notation, the bounds are expressed using symbols that explain whether or not the endpoint is included in the set.

Open and Closed Intervals

In interval notation, whether an endpoint is included in the set defines it as an open or closed interval. This is marked with parentheses ( ) or brackets [ ]:

• **Open Intervals**: Use parentheses, signifying the endpoint is not included (e.g., \((a, b)\)).
• **Closed Intervals**: Use brackets, indicating the endpoint is included (e.g., \([a, b]\)).

Given the inequality \( x < 2 \), we exclusively use open intervals because:

• The number 2 is not included in the interval, so it is marked with a parenthesis \((2)\).
• Negative infinity \((-\infty)\) is not a real number and is always expressed with a parenthesis.

The complete interval notation for \( x < 2 \) is \((-\infty, 2)\), showing that the interval is open at both ends. Understanding these symbols is essential for correctly representing and interpreting intervals.