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

Verified step by step guidance

1. Identify the type of interval: The interval (1, 6] is a half-open interval, which means it includes all numbers greater than 1 and up to and including 6.
2. Express the interval in set-builder notation: In set-builder notation, this interval can be written as \( \{ x \mid 1 < x \leq 6 \} \). This reads as 'the set of all \( x \) such that \( x \) is greater than 1 and less than or equal to 6.'
3. Graph the interval on a number line: To graph this interval, draw a number line and place an open circle at 1 to indicate that 1 is not included in the interval. Then, place a closed circle at 6 to indicate that 6 is included. Shade the region between 1 and 6 to represent all the numbers in the interval.
4. Review the characteristics of the interval: Remember that an open circle on a number line means the endpoint is not included, while a closed circle means the endpoint is included.
5. Check your work: Ensure that the set-builder notation and the graph accurately represent the interval (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.