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Chapter 0: Problem 2

Rewrite in interval notation and graph. $$-4 \leq x \leq-1$$

Short Answer

Expert verified
The interval in notation is \([-4, -1]\) and it includes \(-4\) and \(-1\).

Step by step solution

01

Understand the Inequality

The given inequality is \(-4 \leq x \leq -1\). It states that \(x\) can be any real number between \(-4\) and \(-1\), including \(-4\) and \(-1\) themselves.
02

Rewrite as Interval Notation

In interval notation, we express this range as \([-4, -1]\). The brackets \([\,]\) indicate that the endpoints \(-4\) and \(-1\) are included in the interval.
03

Graph the Interval on a Number Line

To graph \([-4, -1]\) on a number line, draw a solid line segment between \(-4\) and \(-1\). Place a solid dot or a closed circle on \(-4\) and \(-1\) to indicate these numbers are included in the interval.

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Key Concepts

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

Understanding Inequality
An inequality is a mathematical statement that compares two values using symbols such as \(<, \leq, >, \geq\). These symbols indicate the relationship between the values:
  • \(<\): Less than
  • \(\leq\): Less than or equal to
  • \(>\): Greater than
  • \(\geq\): Greater than or equal to
In the inequality \(-4 \leq x \leq -1\), the symbol \(\leq\) means "less than or equal to." Therefore, \(x\) can be any number from \(-4\) to \(-1\), including the endpoints themselves. This kind of inequality is known as a **compound inequality** because it combines two inequalities into one.
Writing inequalities correctly helps in solving mathematical problems and is essential for understanding more complex algebraic concepts. It defines the range or set of values that the variable \(x\) can take.
Creating a Number Line
A number line is a visual representation of numbers placed along a straight line. It's one of the simplest ways to portray the range of values a variable can take, particularly when dealing with inequalities.
To create a number line:
  • Draw a horizontal line and mark equal intervals on it to represent numbers.
  • Choose a suitable scale, so all numbers of interest (like \(-4\) and \(-1\) in our example) are easily located on the line.
  • Label each interval with numbers as needed.
A number line helps you easily see the ordering and spacing of numbers, making it an excellent tool for visualizing inequalities and the values a variable may take. Moreover, it can visually portray concepts like greater than or less than and help in graphing as well.
Graphing Intervals on a Number Line
Graphing intervals on a number line helps you visually represent the set of numbers described by an interval. Let's look at how to graph he interval \([-4, -1]\) from our exercise:
  • First, locate the points \(-4\) and \(-1\) on the number line.
  • Since the interval \([-4, -1]\) includes \(-4\) and \(-1\), place closed or solid dots at these points to signify inclusion.
  • Draw a solid line between them to show that all numbers in this range are part of the solution set.
Using solid lines and points clearly indicates inclusive inequalities. This method of graphing intervals allows one to easily see which numbers are solutions to the inequality.

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