In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. (1, 6]

Key Concepts

Interval Notation

Interval notation is a mathematical notation used to represent a range of numbers. It uses parentheses and brackets to indicate whether endpoints are included or excluded. For example, (1, 6] means that 1 is not included in the interval, while 6 is included. This notation is essential for understanding the boundaries of the interval.

Set-Builder Notation

Set-builder notation is a concise way to express a set by specifying a property that its members must satisfy. For the interval (1, 6], it can be expressed as {x | 1 < x ≤ 6}, meaning 'the set of all x such that x is greater than 1 and less than or equal to 6.' This notation is useful for defining intervals in a more formal mathematical context.

Graphing Intervals on a Number Line

Graphing intervals on a number line visually represents the range of values included in the interval. For (1, 6], you would draw an open circle at 1 (indicating it is not included) and a closed circle at 6 (indicating it is included), then shade the region between them. This graphical representation helps in understanding the interval's limits and the values it encompasses.