Problem 48 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$(\sqrt{2}, \infty)$$

Step-by-Step Solution

Answer

The inequality is \( x > \sqrt{2} \), and on a number line, use an open circle at \( \sqrt{2} \) extending right.

Step 1: Understanding Interval Notation

The interval \( (\sqrt{2}, \infty) \) represents the set of all real numbers greater than \( \sqrt{2} \) but not including \( \sqrt{2} \) itself. The parenthesis indicates that the endpoint is not included.

Step 2: Convert to Inequality Notation

In inequality notation, the given interval is expressed as \( x > \sqrt{2} \). This indicates that any number larger than \( \sqrt{2} \) is part of the interval.

Step 3: Drawing the Number Line

To represent \( (\sqrt{2}, \infty) \) on a number line, draw a line and put an open circle at \( \sqrt{2} \) to show that it is not included. Then draw a line extending to the right towards infinity to represent all numbers greater than \( \sqrt{2} \).

Key Concepts

Interval Notation

Interval notation is a way to describe a set of numbers between two endpoints. It uses brackets and parentheses to define the endpoints, and indicates whether the endpoints are included in the set.

- **Brackets**: [ ] are used when the endpoint is included in the interval. These are called 'closed intervals'. For example, you'd write \([2, 5]\) for the set of real numbers between 2 and 5, including 2 and 5 themselves. - **Parentheses**: ( ) indicate that the endpoint is not included in the interval. These are called 'open intervals'. For instance, \((2, 5)\) covers all real numbers greater than 2 and fewer than 5, but not including these actual numbers. - **Combination of brackets and parentheses**: For intervals including one endpoint and excluding the other, you use a combination of both. For example, \([2, 5)\) encompasses numbers from 2 up to, but not including, 5. When dealing with infinity, always use a parenthesis because infinity is not a number that can be reached, thus it cannot be included.

Real Numbers

Real numbers are all numbers that can be found on the number line. They include all the rational numbers, such as integers and fractions, and all the irrational numbers, numbers that cannot be written as a fraction.

• **Rational numbers**: These can be expressed as a fraction of two integers. For example, 1, 0.5, and -3/4 are rational numbers.
• **Irrational numbers**: These numbers cannot be precisely written as a fraction. Common examples include \( \sqrt{2} \), \( \pi \), and \( e \). These numbers have decimal expansions that do not repeat or terminate.

Real numbers can be plotted on a number line and help represent continuous data sets such as distance, time, etc. Real numbers are a crucial part of understanding mathematics because they form the backbone of almost all numerical concepts taught in math.

Number Line Representation

A number line is a visual representation of numbers in order on a straight line. It helps illustrate the position of numbers in relation to each other. Each point on the number line corresponds to a unique real number.
When representing intervals on a number line, it's important to understand how to indicate inclusivity or exclusivity at endpoints.

• **Open circles**: Used to indicate numbers that are not included in the interval. For example, in the interval \((\sqrt{2}, \infty)\), an open circle at \(\sqrt{2}\) shows that \(\sqrt{2}\) is not part of the interval.
• **Closed dots**: Represent numbers that are included in the interval, like drawing a closed dot to show the endpoint is inclusive.

The line extending from these points on the number line indicates all numbers that are included within the interval. In our example, drawing a line from an open circle at \(\sqrt{2}\) to the right towards infinity visually captures all numbers greater than \(\sqrt{2}\). This method helps in understanding where numbers fall within intervals and makes abstract concepts more tangible.