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Chapter 0: Problem 15

Write interval notation for each of the following. Then graph the interval on a number line. $$\\{x | x \leq-2\\}$$

Short Answer

Expert verified
Interval notation: \((-\infty, -2]\). Graph: A number line with a closed dot at -2 and shading to the left.

Step by step solution

01

Understand the Set Notation

The set notation \(\{x | x \le -2\}\) represents all values of \(x\) that are less than or equal to -2.
02

Write in Interval Notation

To convert the set notation to interval notation, identify the bounds. Since \(x\) can be any value less than or equal to -2, the interval starts from \(-\infty\) and ends at \(-2\). Therefore, the interval notation is \((-\infty, -2]\).\
03

Graph the Interval on a Number Line

Draw a number line, mark -2, and shade the region to the left of -2 to represent all the values less than -2. Since \(x\) is also equal to -2, place a closed dot (or filled circle) on -2.

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Key Concepts

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

set notation
Set notation is a way of describing a collection of objects or numbers. It is often used to indicate all the elements that satisfy a certain property. For example, the set notation \{x \| x \le-2\} reads as 'the set of all x such that x is less than or equal to -2'. This is a concise way to list all possible values visually or mathematically, without having to enumerate each element.

Here are key elements of set notation:
  • Curly Braces \(\{ \} \): Used to define the set itself.
  • Vertical bar \(\|\) or Colon \(:\): Means 'such that,' introducing the condition or property.
  • Element(s) \(x\): What is being described, typically a variable or number.
  • Condition: What the elements must satisfy (e.g., \(x \le -2\)).
Understanding set notation is crucial for converting it into interval notation or inequality form, which are more visual representations.
inequality
Inequalities express a relationship where two values are not necessarily equal. They are useful to describe ranges of possible values rather than a specific number. In an inequality like \(x \le -2\), we're stating that x can be any number less than or equal to -2.
There are a few types of inequalities:
  • Less than (\(<\)): Indicates that one value is smaller than another.
  • Greater than (\(>\)): Indicates that one value is larger than another.
  • Less than or equal to (\(\le\)): Indicates that one value is smaller than or equal to another.
  • Greater than or equal to (\(\ge\)): Indicates that one value is larger than or equal to another.
Inequalities can be represented on number lines or converted into interval notation for a clearer understanding.

In the given problem, the inequality \(x \le -2\) was converted to interval notation \((-\infty, -2]\). This not only expresses the same condition but also makes it easier to visualize.
graphing intervals
Graphing an interval on a number line helps to visualize the solution set of an inequality. To do this, you need to:
  • Draw a Number Line: Start with a horizontal line and mark relevant points.
  • Identify Boundary Points: Mark the specific bounds of the interval (e.g., -2).
  • Shade the Region: For all values that satisfy the inequality. For \(x \le -2\), shade everything to the left of -2.
  • Use Open or Closed Dots: Use a closed dot for \(\le\) or \(\ge\) to include the boundary. Use an open dot for \(<\) or \(>\) to exclude the boundary.
In our problem, draw a number line, locate -2, and place a closed dot there because \(x\) can be equal to -2. Then, shade all the numbers to its left to represent \((-\infty, -2]\). This practice makes abstract concepts more concrete.

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