Problem 27 - Elementary Algebra

Question

Find the distance between the given numbers on a number line. -8 and 14

Step-by-Step Solution

Answer

The distance between -8 and 14 on a number line is 22.

Step 1: Identify the numbers

The two numbers given in the problem are -8 and 14. We need to find the distance between these two numbers on a number line.

Step 2: Understand the distance formula

The distance between two points (or numbers) on a number line is given by the absolute difference of those numbers. This can be written as \(|a - b|\), where \(a\) and \(b\) are the two numbers.

Step 3: Apply the distance formula

Using the distance formula, we calculate the distance as: \[|-8 - 14| = |-8 - 14| = |-22| = 22\]

Step 4: Interpret the result

The calculation shows that the distance between the numbers -8 and 14 on the number line is 22.

Key Concepts

Understanding the Number Line

To visualize numbers and their distances, we often use a number line. Imagine it as a straight, horizontal line where each point represents an integer.

• Positive numbers are placed to the right of zero.
• Negative numbers lie to the left of zero.

To place a number, simply move right or left from zero, counting by ones. For example, -8 would be eight spaces to the left, and 14 would be fourteen spaces to the right. The number line gives us a clear visual representation of distance between numbers. It helps to see how far one number is from another. Even without calculating, you can quickly grasp that numbers like -8 and 14 are very far apart, as they are on opposite sides of zero.

Distance Formula between Two Points

Finding the distance between numbers on a number line involves understanding the distance formula. Simply put, the distance is the absolute difference between two numbers, represented as \( |a - b| \). The absolute value ensures we measure only how far apart numbers are, without worrying about direction or sign.

• The formula is \( |a - b| \), where \(a\) and \(b\) are the numbers.
• "Absolute" means we only consider the positive distance, or simply put, the gap between the numbers.

For example, if the numbers are -8 and 14, the formula becomes \( |-8 - 14| = |-22| = 22 \). This shows the amount of space, or units, between these numbers. No matter their placement on the line, this formula gives a straightforward solution to find their distance.

The Role of Integer Subtraction

Integer subtraction is often a part of calculating distance on a number line. When finding the distance between two integers, \(a\) and \(b\), subtraction tells us how far apart they are. Performing \(a - b\) indicates the numerical difference, which is then converted to an absolute value to give us the distance.

• Integer subtraction helps find the difference between any two numbers.
• The result can be positive or negative, but with absolute value, the sign does not matter.

In our example, subtracting 14 from -8 results in \(-22\), reflecting both the direction (to the left on a number line) and difference in magnitude. Taking the absolute value, \( |-22| \), gives the final distance of 22. This process highlights how subtraction relates to measuring real distances on the number line.