Problem 82 - Precalculus Student Solutions Manual 5th

Question

Write an equation involving absolute value that says the distance between \(r\) and \(s\) is 6 units.

Step-by-Step Solution

Answer

\[ |r - s| = 6 \]

Step 1 - Understand the Problem

The absolute value of the difference between two numbers represents the distance between them. For this problem, the distance between two numbers, denoted by the absolute value, must be 6.

Step 2 - Set up the Absolute Value Equation

Let the two numbers be represented by variables, say, \(r\) and \(s\). We need an equation that involves the absolute value of the difference between \(r\) and \(s\). This can be written as \(|r-s|\) or \(|s-r|\).

Step 3 - Write the Equation

Since the distance between \(r\) and \(s\) is 6 units, we can write: \[ |r - s| = 6 \]

Key Concepts

Absolute Value Difference

Absolute value is a mathematical function that's used to describe the magnitude, or size, of a number without regard to its direction (positive or negative). In simpler terms, it converts all numbers to their positive counterparts.
Suppose you have two numbers, let's call them \( r \) and \( s \). The absolute value of the difference between these two numbers, expressed as \( |r - s| \) or \( |s - r| \), represents the distance between them on the number line.
For example, if \( r = 3 \) and \( s = 8 \), then \( |3 - 8| \) equals \( |-5| \), which is 5. Conversely, if \( r = 8 \) and \( s = 3 \), then \( |8 - 3| \) equals 5. So, \(|r - s| = 5 \) in both cases.
When the problem states that the distance between \( r \) and \( s \) is 6 units, what it really means is that \( |r - s| = 6 \). This is why we use the absolute value function in the equation.

Setting Up Equations

Setting up equations is a fundamental skill in algebra. This involves translating a word problem or concept into a mathematical expression.
Let's break it down: we need to write an equation that says the distance between two numbers, \( r \) and \( s \), is 6 units. Based on what we've covered, we know that distance is given by the absolute value of the difference between these numbers, which is \( |r - s| \).
Since the distance is 6, we can set this up as:

• \( |r - s| = 6 \)

This equation tells us that the difference between the numbers \( r \) and \( s \) is either 6 or -6. If we solve for both scenarios, we get two equations: \( r - s = 6 \) and \( r - s = -6 \).
In summary, setting up equations based on absolute value differences can help us determine all possible solutions to problems involving distance between numbers.