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Chapter 0: Problem 5

Express each interval in set-builder notation and graph the interval on a number line. $$[-3,1]$$

Short Answer

Expert verified
The set-builder notation for the interval \([-3,1]\) is \({x|-3\leq x\leq 1}\)

Step by step solution

01

Identify the Interval

The given interval is \([-3,1]\). This means x is in the interval from -3 to 1 inclusive. This is a closed interval, which means it includes the endpoints.
02

Convert to Set-builder notation

The set-builder notation of a given interval is defined as \({x|-3\leq x\leq 1}\). This can be read as 'the set of all x such that x is greater than or equal to -3 and less than or equal to 1'. This implies all the real numbers between -3 and 1 inclusive.
03

Graph the Interval

To represent this on a number line, mark -3 and 1 and darken the line between them to indicate the interval. As the interval is closed at both ends (-3 and 1 are included in the set), small darkened circles are drawn at these points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set-builder notation
Set-builder notation is a concise way of representing a set by specifying a property that its members must satisfy. It is widely used in mathematics to define intervals of numbers. For instance, the set-builder notation for the interval
  • Represents a set by describing the properties of its elements.
  • Uses a variable, such as \(x\), and conditions to define the set.
  • An example is \(\{x \mid -3 \leq x \leq 1\}\), which reads as "the set of all x such that \(-3\) is less than or equal to \(x\) and \(x\) is less than or equal to \(1\)."
This may seem complex, but it simply categorizes all numbers between two given points, including those points. This notation is ideal for clarifying which parts of the number line are included in the set.
Closed interval
A closed interval is an interval that includes its endpoints. In mathematical notation, the closed interval from \(a\) to \(b\) is represented as
  • Indicated by square brackets: \([a, b]\).
  • Includes all numbers \(x\) such that \(a \leq x \leq b\).
  • The endpoints are part of the set, unlike open intervals which use parentheses (e.g., \((a, b)\)) to exclude endpoints.
For our example, \([-3, 1]\) means every real number from \(-3\) to \(1\), inclusive. This is important when graphing, as closed intervals require you to show the endpoints are included, often by using filled-in circles on a number line to visually indicate that endpoints are part of the interval.
Number line
Number lines provide a straightforward way to visualize and understand intervals. It's a horizontal line with numbers placed at equal intervals, typically centering on zero. To graph a closed interval on a number line:
  • Identify the endpoints of the interval. For \([-3, 1]\), these are \(-3\) and \(1\).
  • Darken or draw a line connecting these two points to show the range of the interval.
  • Fill in the circles at the endpoints to indicate they are included in the interval.
Using a number line helps students easily see which parts of the line (or which numbers) are part of the interval. It provides a clear, visual representation of an otherwise abstract mathematical idea.
Inequalities
Inequalities are mathematical expressions involving a relational operator, explaining how one number compares to another. There are various types of inequalities that determine the inclusion or exclusion of numbers. The key aspects include:
  • Use symbols such as \(\leq\) (less than or equal to) and \(\geq\) (greater than or equal to).
  • For the closed interval \([-3, 1]\), the inequality \(-3 \leq x \leq 1\) indicates \(x\) can be any number between \(-3\) and \(1\), inclusive.
  • Inequalities help us describe an interval's boundaries mathematically.
Understanding inequalities is crucial for solving and interpreting solutions to algebraic problems, as they define the limits within which solutions must fall. By mastering inequalities, students can effectively determine the set of numbers that satisfy a given condition.

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