In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. [- 3, 1]

Verified step by step guidance

1. <Step 1: Understand the interval notation [-3, 1]. This notation represents all the numbers between -3 and 1, including -3 and 1 themselves.>
2. <Step 2: Convert the interval into set-builder notation. Set-builder notation describes a set by a property that its members must satisfy. For this interval, it can be expressed as: \( \{ x \mid -3 \leq x \leq 1 \} \).>
3. <Step 3: Prepare to graph the interval on a number line. Identify the endpoints of the interval, which are -3 and 1.>
4. <Step 4: On the number line, mark the points -3 and 1 with solid dots to indicate that these endpoints are included in the interval.>
5. <Step 5: Draw a solid line connecting the points -3 and 1 on the number line to represent all the numbers between and including -3 and 1.>

Key Concepts

Set-Builder Notation

Set-builder notation is a mathematical shorthand used to describe a set by specifying a property that its members must satisfy. For example, the interval [-3, 1] can be expressed in set-builder notation as {x | -3 ≤ x ≤ 1}, meaning 'the set of all x such that x is greater than or equal to -3 and less than or equal to 1.' This notation is particularly useful for defining intervals and sets in a concise manner.

Intervals

An interval is a range of numbers between two endpoints, which can be open, closed, or half-open. A closed interval, like [-3, 1], includes its endpoints, meaning both -3 and 1 are part of the set. Understanding the types of intervals is crucial for accurately expressing them in set-builder notation and for graphing them on a number line.

Graphing on a Number Line

Graphing an interval on a number line involves visually representing the range of values included in the interval. For the closed interval [-3, 1], you would draw a solid dot at -3 and 1 to indicate that these endpoints are included, and shade the region between them. This visual representation helps in understanding the extent of the interval and the values it encompasses.