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

Verified step by step guidance

1. Identify the type of interval given: \([-3, \infty)\) is a half-open interval.
2. In set-builder notation, express the interval as \( \{ x \mid x \geq -3 \} \).
3. Understand that \([-3, \infty)\) includes all real numbers greater than or equal to \(-3\).
4. To graph the interval on a number line, draw a solid dot at \(-3\) to indicate that \(-3\) is included in the interval.
5. Draw a line extending to the right from \(-3\) towards infinity, indicating that all numbers greater than \(-3\) are included.

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, ∞) can be expressed in set-builder notation as {x | x ≥ -3}, indicating that the set includes all real numbers x that are greater than or equal to -3.

Intervals

An interval is a range of numbers between two endpoints. In this case, the interval [-3, ∞) includes all numbers starting from -3 and extending indefinitely to the right. The square bracket indicates that -3 is included in the interval, while the infinity symbol (∞) signifies that there is no upper limit.

Graphing on a Number Line

Graphing an interval on a number line involves visually representing the range of values included in the interval. For [-3, ∞), you would place a closed dot at -3 to indicate it is included, and then shade the line to the right to show that all numbers greater than -3 are part of the interval, extending indefinitely.