In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. (2, ∞)

Verified step by step guidance

1. Identify the type of interval given: (2, ∞) is an open interval starting from 2 and extending to infinity.
2. In set-builder notation, express the interval as \( \{ x \mid x > 2 \} \). This means the set of all \( x \) such that \( x \) is greater than 2.
3. To graph the interval on a number line, draw a number line with a point at 2.
4. Use an open circle at 2 to indicate that 2 is not included in the interval.
5. Draw an arrow extending to the right from the open circle to represent all numbers greater than 2, continuing indefinitely.

Key Concepts

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included or excluded. For example, (2, ∞) means all numbers greater than 2, where 2 is not included, and ∞ indicates that there is no upper limit.

Set-Builder Notation

Set-builder notation is a concise way to describe a set by specifying a property that its members must satisfy. For the interval (2, ∞), the set-builder notation would be {x | x > 2}, meaning 'the set of all x such that x is greater than 2.' This notation is particularly useful for defining infinite sets.

Graphing Intervals

Graphing intervals on a number line visually represents the range of values included in the interval. For (2, ∞), you would draw an open circle at 2 (indicating that 2 is not included) and shade the line to the right towards infinity, illustrating that all numbers greater than 2 are part of the interval.