In Exercises 104–106, express each interval in set-builder notation and graph the interval on a number line. (-2, ∞)

Verified step by step guidance

1. Understand the given interval: The interval (-2, ∞) represents all real numbers greater than -2 but not including -2 itself. The parentheses indicate that -2 is not part of the interval, and ∞ means the interval extends indefinitely in the positive direction.
2. Convert the interval to set-builder notation: In set-builder notation, we describe the set of numbers using a condition. For this interval, the condition is that x is greater than -2. This can be written as {x | x > -2}, where the vertical bar '|' means 'such that.'
3. Prepare to graph the interval on a number line: To graph the interval (-2, ∞), start by marking the point -2 on the number line. Since -2 is not included in the interval, use an open circle at -2 to indicate this.
4. Draw the graph: From the open circle at -2, draw a line extending to the right indefinitely to represent all numbers greater than -2. Add an arrow at the end of the line to show that the interval continues infinitely.
5. Review the solution: The interval (-2, ∞) is expressed in set-builder notation as {x | x > -2}, and the graph includes an open circle at -2 with a line extending to the right indefinitely.

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 (-2, ∞) can be expressed in set-builder notation as {x | x > -2}, meaning 'the set of all x such that x is greater than -2'. 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. Intervals can be open, closed, or half-open, depending on whether the endpoints are included. The interval (-2, ∞) is an open interval starting at -2 and extending indefinitely to the right, meaning -2 is not included in the set, but all numbers greater than -2 are.

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 interval (-2, ∞), you would place an open circle at -2 to indicate that it is not included, and then shade the line to the right to show that all numbers greater than -2 are part of the interval. This visual representation helps in understanding the extent of the interval.