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. Identify the type of interval: The interval (-2, ∞) is an open interval on the left at -2 and extends indefinitely to the right.
2. Write the interval in set-builder notation: The set-builder notation for this interval is \{x | x > -2\}. This notation reads as 'the set of all x such that x is greater than -2'.
3. Draw a number line: Start by drawing a horizontal line and marking points on it. Be sure to include the point -2.
4. Mark the interval on the number line: Since -2 is not included in the interval (indicated by the parenthesis), draw an open circle at -2. Then, shade the line to the right of -2 to indicate that all numbers greater than -2 are included in the interval.
5. Extend the shaded region to the right indefinitely: Since the interval extends to infinity, draw an arrow at the end of the shaded region pointing to the right to indicate that it continues indefinitely.

Key Concepts

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included or excluded. For example, the interval (-2, ∞) includes all numbers greater than -2 but does not include -2 itself, as indicated by the parentheses.

Set-Builder Notation

Set-builder notation is a concise way to express a set by specifying a property that its members must satisfy. It typically takes the form {x | condition}, where 'x' represents the elements of the set and 'condition' describes the criteria for membership. For the interval (-2, ∞), the set-builder notation would be {x | x > -2}, indicating all x values greater than -2.

Graphing Intervals on a Number Line

Graphing intervals on a number line visually represents the range of values included in the interval. Open intervals, indicated by parentheses, are represented with an open circle at the endpoint, showing that the endpoint is not included. For (-2, ∞), you would place an open circle at -2 and shade the line to the right, indicating all values greater than -2.