In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. (2, ∞)

Verified step by step guidance

1. Identify the interval given: (2, \infty). This interval includes all real numbers greater than 2 but does not include 2 itself.
2. Recall that in set-builder notation, we describe the set of all elements x that satisfy a certain condition. Here, the condition is that x is greater than 2.
3. Write the set-builder notation as: \(\{ x \mid x > 2 \}\), which reads as "the set of all x such that x is greater than 2."
4. To graph this interval on a number line, draw a number line and mark the point 2 with an open circle to indicate that 2 is not included in the interval.
5. Shade the number line to the right of 2 extending towards positive infinity to represent all numbers greater than 2.

Key Concepts

Interval Notation

Interval notation is a way to represent a set of numbers between two endpoints. For example, (2, ∞) means all numbers greater than 2 but not including 2, extending to infinity. Parentheses indicate that endpoints are not included, while brackets indicate inclusion.

Set-Builder Notation

Set-builder notation describes a set by specifying a property that its members must satisfy. For intervals, it typically uses a variable and an inequality, such as {x | x > 2}, meaning the set of all x such that x is greater than 2.

Graphing Intervals on a Number Line

Graphing an interval involves shading the portion of the number line that represents the set. Open circles indicate endpoints not included (like 2 in (2, ∞)), and shading extends towards infinity to show all numbers greater than the endpoint.