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Chapter 1: Problem 44

Express the interval in terms of inequalities, and then graph the interval. $$[-6,-1]$$

Short Answer

Expert verified
The interval \([-6,-1]\) as inequalities is \(-6 \leq x \leq -1\), and it includes all numbers between \(-6\) and \(-1\) on a number line.

Step by step solution

01

Understand the Closed Interval

The interval \([-6, -1]\) is a closed interval. This means it includes both endpoints, \(-6\) and \(-1\). In interval notation, brackets \([ \text{and} ]\) indicate that the endpoints are included.
02

Convert the Interval to Inequalities

Express the closed interval \([-6, -1]\) using inequalities. It can be written as \(-6 \leq x \leq -1\). This inequality states that \(x\) is greater than or equal to \(-6\) and less than or equal to \(-1\).
03

Graph the Interval on a Number Line

To graph the interval \([-6, -1]\), draw a number line and mark the points \(-6\) and \(-1\). Since the interval is closed, use solid dots or circles at \(-6\) and \(-1\) to indicate that these values are included in the interval. Draw a line connecting \(-6\) and \(-1\) to show all numbers \(x\) such that \(-6 \leq x \leq -1\).

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Key Concepts

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

Closed Interval
When you see the notation \([-6, -1]\), you're looking at what mathematicians call a "closed interval." Understanding intervals is important for grasping how ranges of numbers are described in mathematics.
In this case, a closed interval means that the values at both ends of the interval, which are - the start, at -6 - and the end, at -1
are included in the set of numbers described by the interval.
  • The square brackets, \([ \, ]\), tell us the endpoints are included.
  • This is different from open intervals, \((\ \)) where endpoints are not included.
Understanding the distinction between closed and open intervals is foundational for working with ranges of numbers whether solving equations or plotting data.
Inequalities
Inequalities are expressions that explain the relationship between two values or expressions. They tell us how one value compares in size to another and often use symbols such as:
  • \(<\) which means "less than"
  • \(>\) which means "greater than"
  • \(\leq\), less than or equal to
  • \(\geq\), greater than or equal to

When expressing the closed interval \([-6, -1]\) in terms of inequalities, you write it as \(-6 \leq x \leq -1\).
This describes the range of values that \(x\) can take.
It tells us:- \(x\) is a number that can be as small as -6, or equal to -6, and- as big as -1, or equal to -1.
Inequalities are powerful tools in mathematics, allowing us to express not just single values, but entire ranges of possibilities.
Number Line Graph
Graphing intervals on a number line is a visual way to show all the numbers that fit within a certain range. For the interval \([-6, -1]\), graphing it on a number line helps solidify its meaning.
Here's how to do it:
  • Draw a horizontal line, which represents all real numbers.
  • Mark points for -6 and -1 on the line. These are boundaries of the interval.
  • Because it is a closed interval, use solid dots or filled circles to show that -6 and -1 are part of the interval.
  • Draw a line or bold line segment connecting these points to illustrate all the values between -6 and -1.
Using these dots and lines effectively can make the concept of an interval much easier to understand. It's a helpful method to visually express and interpret mathematical ideas, showing clearly which values are part of a solution and which are not.

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Most popular questions from this chapter

Show that the equation represents a circle, and find the center and radius of the circle. $$x^{2}+y^{2}-\frac{1}{2} x+\frac{1}{2} y=\frac{1}{8}$$

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State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$\frac{-a}{b}=-\frac{a}{b}$$

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