In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. [- 5, 2)

Verified step by step guidance

1. Identify the type of interval given. The interval is \([-5, 2)\), which means it includes all real numbers from \(-5\) to \(2\), including \(-5\) but excluding \(2\).
2. Recall that in set-builder notation, we describe the set of all \(x\) such that \(x\) satisfies certain inequalities. For this interval, \(x\) must be greater than or equal to \(-5\) and less than \(2\).
3. Write the inequality form for the interval: \(-5 \leq x < 2\).
4. Express the interval in set-builder notation as: \(\{ x \mid -5 \leq x < 2 \}\), which reads as "the set of all \(x\) such that \(x\) is greater than or equal to \(-5\) and less than \(2\)."
5. To graph the interval on a number line, draw a solid dot at \(-5\) to indicate it is included, and an open circle at \(2\) to indicate it is not included. Shade the region between \(-5\) and \(2\) to represent all numbers in the interval.

Key Concepts

Interval Notation

Interval notation is a way to represent a set of numbers between two endpoints. Square brackets [ ] indicate that an endpoint is included (closed interval), while parentheses ( ) mean the endpoint is excluded (open interval). For example, [-5, 2) includes -5 but excludes 2.

Set-Builder Notation

Set-builder notation describes a set by specifying a property that its members satisfy. For intervals, it typically uses inequalities, such as {x | -5 ≤ x < 2}, meaning the set of all x such that x is greater than or equal to -5 and less than 2.

Graphing Intervals on a Number Line

Graphing intervals involves marking the endpoints on a number line and shading the region between them. Closed endpoints are shown with solid dots, indicating inclusion, while open endpoints use hollow dots, indicating exclusion. This visualizes the set of numbers in the interval.