Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Find the distance between \(a\) and \(b\).\(a=126, b=75\)

Step-by-Step Solution

Answer

The distance between \(a\) and \(b\) is 51.

Step 1: Identify given numbers

The numbers given are \(a = 126\) and \(b = 75\).

Step 2: Apply the formula for distance

The mathematical formula to find the distance between any two points \(a\) and \(b\) on a number line is by calculating the absolute value of their difference, given as \(|a - b|\).

Step 3: Perform the calculation

Calculate \(|a - b|\) = \(|126 - 75|\), which simplifies to \(|51|\).

Key Concepts

Absolute Value

Absolute value is a concept often used in mathematics to determine the distance of a number from zero on the number line. It signifies how far a number is from zero, regardless of direction. To find the absolute value, we denote it using vertical bars like this: \(|x|\).
- If the number inside is positive or zero, the absolute value is the number itself.- If the number inside is negative, the absolute value is the number without the negative sign.
For example, \(|5| = 5\) and \(|-5| = 5\). Notice that both values are 5 units away from zero on the number line. Absolute value is often used in distance calculations, such as finding the difference between two numbers like \(|a - b|\). This is crucial in distance calculations as it always returns a non-negative result, ensuring a true measurement of distance.

Number Line

The number line is a straight, horizontal line that is used to represent numbers. Each point on the line corresponds to a number, with zero typically placed at the center. Numbers to the right of zero are positive, whereas numbers to the left are negative. This visualization helps in understanding the concept of distance in mathematics, making it easier to grasp how far apart numbers are.
To find the distance between two points (numbers) on the number line:

• Identify the two numbers in question, such as 126 and 75.
• Visualize their positions on the number line.
• Calculate their distance by finding the absolute value of their difference.

This method provides a visual and tangible way to understand that distance reflects the space between numbers, irrespective of direction. It simplifies problems involving numerical differences, enabling learners to see the practicality of the absolute value in measuring real distances.

Mathematical Formula

The concept of a mathematical formula is crucial in efficiently solving problems by applying a standardized method. For finding the distance between two points on a number line, the formula utilized is \(|a - b|\). This formula is elegant and effective for computing the distance because it integrates the idea of absolute value directly into the calculation.
Let's break it down step-by-step:- Identify the two numbers, which would be our points, such as \(a = 126\) and \(b = 75\).- Compute the difference: subtract the smaller number from the larger one, or vice versa, to get the point difference, ensuring to handle order appropriately.- Apply the absolute value: \(|126 - 75|\) simplifies to \(|51|\).- Result: since the absolute value of 51 is 51, the computed distance is 51.
This formula guarantees that the distance is treated consistently as a non-negative value, making it especially helpful when precise distance measurements are crucial, such as in real-world applications like navigation or geometry.