Problem 66 - College Algebra and Calculus: An Applied Approach

Question

Find the distance between \(a\) and \(b\).\(a=\frac{3}{4}, b=\frac{9}{4}\)

Step-by-Step Solution

Answer

The distance between the two points \(a\) and \(b\) is \(1.5\) or \(\frac{3}{2}\).

Step 1: Identify points on the number line

Firstly, we have the two points given to us as fractions: \(a=\frac{3}{4}\) and \(b=\frac{9}{4}\)

Step 2: Calculate the difference

Then, the distance between \(a\) and \(b\) is calculated as the absolute value of the difference \(|a-b|\). So, we substitute values of \(a\) and \(b\) into the equation: \(|\frac{3}{4}-\frac{9}{4}|\)

Step 3: Simplify

Simplify the fraction within the absolute value sign. You have \(-\frac{6}{4}\), which simplifies to \(-\frac{3}{2}\). Since the distance cannot be negative, the absolute value changes the negative difference into positive.

Key Concepts

Absolute Value

Absolute value is a concept in mathematics that tells us how far a number is from zero without considering its direction. It's always a non-negative number. For example, the absolute value of 5 is 5, and the absolute value of -5 is 5 too.
When we talk about distance, such as finding the distance between two points, we use the absolute value to ensure the result is positive.

• Consider the numbers on a number line. Their distance is the absolute value of the difference between them.
• This is written as \(|a - b|\), where "a" and "b" are the two points.

Just like in our exercise, where we found the absolute value of \(-\frac{3}{2}\) when simplifying the difference between the fractions \(\frac{3}{4}\) and \(\frac{9}{4}\). No matter the order of subtraction, the absolute value ensures the distance is non-negative.
Understanding absolute value helps make calculating distance straightforward and consistent.

Fractions

Fractions are a way to represent parts of a whole. A fraction is composed of two numbers: the numerator (top number) and the denominator (bottom number). For instance, in \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator.
In our exercise, both points were given as fractions: \(a = \frac{3}{4}\) and \(b = \frac{9}{4}\). To find the distance between these points, we first had to subtract one fraction from the other.

• Make sure the fractions have the same denominator before subtracting.
• If needed, adjust the fractions to equivalent forms with a common denominator.
• Subtract the numerators while keeping the denominator the same.
• Simplify the resulting fraction, if possible.

Once we calculated \(\frac{3}{4} - \frac{9}{4} = -\frac{6}{4}\), we simplified it to \(-\frac{3}{2}\). Fractions allow for precise representation of values between whole numbers, which can be essential for accurate calculations.

Number Line

A number line is a visual representation of numbers in a straight line, where each point corresponds to a number. It's a useful tool for understanding and visualizing concepts such as addition, subtraction, and distance.
The number line typically has zero in the center, with positive numbers extending to the right and negative numbers extending to the left.

• It's ideal for plotting and comparing fractions, decimals, and integers.
• Distances between points on a number line can be found by measuring the gap, which is a perfect application of absolute value.
• Steps on a number line are equally spaced and represent consistent intervals.

In our exercise, plotting \(\frac{3}{4}\) and \(\frac{9}{4}\) on a number line helps visualize their positions and the distance between them. By seeing these positions, it's easier to comprehend why taking their absolute difference represents the actual distance. The number line is a fundamental tool for mathematics that supports many calculations and problem-solving strategies.