Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show the interval on a number line. $$ (-2,5] $$

Step-by-Step Solution

Answer

To represent the interval \((-2, 5]\) on a number line, first draw a horizontal line with consecutive integers marked on it, including -2 and 5. Mark an open circle at -2 (since it's not included) and a closed circle at 5 (since it's included). Connect the two circles with a thick solid line to represent the interval \((-2, 5]\) on the number line.

Step 1: Draw a number line

First, draw a horizontal line, which represents the number line. Mark points at equal distances along the line and label them with consecutive integers, including -2 and 5, as well as a few integers before and after them to give some context.

Step 2: Mark the interval endpoints

On the number line, mark clearly the endpoints of the given interval, which are -2 and 5. Since -2 is not included in the interval, draw an open circle at -2. On the other hand, since 5 is included in the interval, draw a closed-circle (filled) at 5.

Step 3: Draw the interval

Now, draw a thick solid line connecting the open circle at -2 and the closed circle at 5, to represent the interval (-2, 5]. Upon completion of these steps, the number line will successfully represent the given interval (-2, 5].

Key Concepts

Number Line

A number line is a straight, horizontal line that helps us visualize numbers and their relative positions. Imagine it like a ruler that extends infinitely in both directions, filled with numbers placed at equal distances. It simplifies complex mathematical concepts and is incredibly useful for understanding intervals and calculations. When working with a number line, remember to:

• Place zero at the center, marking it as a point of reference.
• Ensure equal spacing between consecutive integers for consistency.
• Label numbers accurately for clear visualization.

A number line serves as a backdrop for representing numbers and understanding their relationships, making it easier to grasp abstract concepts like interval notation.

Open and Closed Circles

When drawing intervals on a number line, you'll often use open and closed circles to indicate which endpoint values are included or excluded. Open and closed circles help us provide a clear visual cue.

**Open Circles**

• An open circle indicates that a value is not part of the interval.
• It is drawn as a hollow circle.
• For example, in the interval \((-2, 5]\), the open circle is at -2, as -2 is not included.

**Closed Circles**

• A closed circle means the value is part of the interval.
• It is represented by a filled, solid circle.
• Within the interval \((-2, 5]\), the number 5 includes a closed circle because it is part of the interval.

Using open and closed circles correctly allows you to distinguish clearly between included and excluded endpoints.

Endpoints

Endpoints are critical when defining an interval, as they set the boundaries and tell us where the interval starts and stops. Understanding the role of endpoints will help you interpret intervals accurately on a number line.

**Identifying Endpoints**

• These are usually numbers given in the interval notation, like -2 and 5 in \((-2, 5]\).
• Mark them clearly on the number line before anything else.

**Including or Excluding Endpoints**

• Use open circles for excluded endpoints, and closed circles for included ones.
• In \((-2, 5]\), mark -2 with an open circle because it's not part of the interval, and 5 with a closed circle because it is included.

The treatment of endpoints determines the nature of the interval, and careful attention to how they are marked ensures a clear, accurate representation of any given interval.