Problem 15 - Prealgebra and Introductory Algebra

Question

On the number line, which number is: 3 units to the right of \(-2 ?\)

Step-by-Step Solution

Answer

The number is 1.

Step 1: Identify the Initial Position

First, find the location of -2 on the number line. This serves as the starting point for the next movement.

Step 2: Move Along the Number Line

Moving right on the number line corresponds to addition. Therefore, to move 3 units to the right from -2, you need to add 3 to -2.

Step 3: Perform the Addition

Adding 3 to -2 results in 1. Therefore, the number 1 is 3 units to the right of -2 on the number line.

Key Concepts

Addition

Addition is a fundamental mathematical operation that combines two numbers into a single sum.
Addition on a number line is visual and intuitive. You can see this as moving from left to right during addition.
Let's explore how this works:

• Start from a number (like \(-2\) in our example), on the number line.
• To "add 3" means to count 3 spaces to the right.

For instance, if we start at \(-2\) and add \(3\), we follow these three steps:

• Find \(-2\) (your starting point).
• Move three steps to the right.
• Stop where you are, which, in this case, is \(1\).

The result is the new number, \(1\), which is highlighted by the direction and distance of travel on the number line.

Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign (e.g., \(-2\)).
Here's how they work:

• On a number line, negative numbers are found to the left of zero.
• They represent values below zero, used often for temperatures, debts, and other 'below zero' quantities.

Using negative numbers in addition requires noticing directional movements:

• Adding positive numbers (like \(3\), in our example) to a negative number (like \(-2\)) involves moving to the right on a number line.
• The resulting number moves towards the positive side or to zero.

Negative numbers are key to finding opposites or indicating reductions.

Coordinate System

A number line is a simple version of a coordinate system, but it only covers one dimension.
In a coordinate system:

• Positions are defined by numbers and show spatial relationships.
• The number line represents all numbers arranged in order of size.

Using a number line, you can identify positions through counting steps or units along the line:

• Find the starting point (like \(-2\)).
• Compute movements by steps to reach a new position (like moving 3 steps right).

This simple structure makes it easier to visualize and understand operations such as addition or subtraction. Coordinate systems allow plotting any point and are foundational for more complex math such as graphing equations or identifying spatial points.