Problem 15 - Elementary and Intermediate Algebra

Question

What set is represented by the interval notation \((-\infty, \infty) ?\) Graph it.

Step-by-Step Solution

Answer

The interval (-∞, ∞) represents all real numbers.

Step 1: Understanding Interval Notation

The interval notation (-∞, ∞) represents all real numbers. It includes every number on the number line from negative infinity to positive infinity.

Step 2: Graphing the Interval

To graph this interval on a number line, draw a horizontal line with arrows at both ends to indicate it extends indefinitely. Since it includes all real numbers, every point on the number line is part of the graph.

Key Concepts

Understanding Interval Notation

Interval notation is a mathematical concept used to describe a range of numbers in a clear and concise manner. It is especially helpful when dealing with sets of real numbers. An interval can be described using two main types of brackets:

• Parentheses, like \(a, b\), indicate that the endpoints are not included in the interval.
• Brackets, like \[a, b\], are used when the endpoints are included.

In our problem, the interval notation \((-\infty, \infty)\) represents all real numbers. This means that the interval starts from negative infinity and goes all the way to positive infinity without any breaks or holes. Since infinity is a concept and not a specific number, it is always represented with parentheses to show that it is not an element of the set. This particular interval basically represents the entire number line.

Using the Number Line

A number line is a visual representation of numbers in a straight horizontal line where each point corresponds to a number. This simple but powerful tool helps in understanding and comparing numbers easily.

For the interval \((-\infty, \infty)\), a number line effectively shows all real numbers. To draw this, you start with a straight horizontal line and place arrows on both ends. These arrows symbolize that the line extends infinitely in both directions. This graphical representation clearly communicates that there are no boundaries or endpoints—which fits perfectly with the concept of all-encompassing real numbers described by \((-\infty, \infty)\).

When using a number line, keep these pointers in mind:

• The further left you move, the smaller the numbers become.
• The further right you move, the larger the numbers become.
• Zero is typically placed in the middle, acting as a reference point.

Graphing Intervals

Graphing intervals helps us visually understand the range of numbers we are dealing with. On a number line, graphing allows us to illustrate which numbers are included in the interval and which are not.

For the interval \((-\infty, \infty)\), graphing it on a number line involves:

• Drawing a horizontal line with arrows on both ends to symbolize that it extends indefinitely.
• Highlighting or shading the whole line to indicate that every real number is part of this interval.

This method clearly shows that the set includes all numbers, without any exclusions.

Graphing intervals can be simplified as:

• Identify the interval bounds and whether they include or exclude endpoints.
• Use parentheses or brackets to denote open or closed intervals respectively.
• Draw and indicate portions of the number line that correspond to the interval.

Using these steps keeps your graphs accurate and easy to interpret, making them a valuable tool in both learning and communicating mathematical ideas.