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Chapter 0: Problem 11

Write interval notation for each of the following. Then graph the interval on a number line. The set of all numbers \(x\) such that \(-2 \leq x \leq 2\)

Short Answer

Expert verified
The interval is d[-2, 2]d.

Step by step solution

01

Identify the inequality

The inequality provided is d eq -2 eg x eg 2d which means that x can take any value between -2 and 2, including -2 and 2.
02

Write in interval notation

Interval notation includes both endpoints and is written as n d eq -2, 2 d
03

Graph the interval on a number line

To graph d eq -2, 2 d on a number line, draw a solid circle at -2 and another solid circle at 2. Then, shade the number line between -2 and 2 to indicate all the numbers within this closed interval.

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Key Concepts

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

Inequality
An inequality is a mathematical expression that shows the relationship between two values that are not necessarily equal. In our exercise, the inequality provided is \( -2 \leq x \leq 2 \). This means that \( x \) can be any number between \(-2 \) and \( 2 \), including \(-2 \) and \( 2 \) themselves. This type of inequality is called a 'closed' inequality because the endpoints are included.

In general, inequalities can be:
  • Strict, where endpoints are not included (e.g., \( x < 2 \))
  • Non-strict, where endpoints are included (e.g., \( x \leq 2 \))
Inequalities help us describe a range of values rather than a specific number. Understanding inequalities is essential for graphing and interpreting data on a number line.
Number Line
A number line is a visual representation of numbers on a straight horizontal line. Each point on the line corresponds to a number.

This helps us see the relationships between numbers and understand intervals and inequalities better.

To graph an interval on a number line, follow these steps:
  • Identify the endpoints. For the interval \([-2, 2] \), the endpoints are \(-2 \) and \(2 \).
  • Draw solid circles at the endpoints if the interval is closed. Draw open circles if the interval is open. In this case, draw solid circles at \(-2 \) and \(2 \).
  • Shade the region between the endpoints. This indicates that all numbers between \(-2 \) and \(2 \), including \(-2 \) and \(2 \), are part of the interval.
Graphing on a number line is a helpful way to visualize the range of values described by an inequality.
Interval
An interval is a set of numbers that lie between two endpoints. It can be represented in different ways such as interval notation, a number line, or an inequality.

For our exercise, the interval \( -2 \leq x \leq 2 \) can be written in interval notation as \([-2, 2] \).

Here are the basic types of intervals:
  • Closed interval \( [a, b] \): Includes endpoints \(a \) and \(b \).
  • Open interval \( (a, b) \): Excludes endpoints \(a \) and \(b \).
  • Half-open interval \( [a, b) \) or \( (a, b] \): Includes one endpoint but excludes the other.
Using interval notation helps us quickly describe the set of values within the specified range. It is a convenient and concise way of expressing an interval in mathematical problems.

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