Problem 70 - College Algebra

Question

Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. \(-6\) and 8

Step-by-Step Solution

Answer

The distance between -6 and 8 is 14 units.

Step 1: Forming the Expression

Form an expression to denote the distance between -6 and 8 using absolute value. This is done by subtracting one number from the other and taking the absolute value. Since a distance is always positive, order doesn't matter here. So, the expression would be \(|\,-6 - 8\,|\) or \(|\,8 - (-6)\,|\).

Step 2: Simplifying the Expression

Simplify the expressions made in step 1, which would give \(|\,-14\,|\) or \(|\,14\,|\). These are equivalent.

Step 3: Evaluating the Absolute Value

Evaluate the simplified absolute value expression. The absolute value of any number is the non-negative value of that number. Hence, the absolute value of -14 or 14 is 14. As positive 14 and negative 14 are both at a 'distance' of 14 units from zero on the number line.

Key Concepts

Evaluating Absolute Value

The process of evaluating absolute value involves transforming a given number into its non-negative counterpart. The term 'absolute' refers to the true magnitude of a number without considering its sign, making the absolute value of both positive and negative numbers the same if they have the same numerical value.

When you come across an expression such as (|-6 - 8|), your first step is to combine the numbers inside the absolute value signs. In this example, -6 minus 8 equals -14. The absolute value sign serves like a 'magic mirror' that reflects any negative number into a positive one. Therefore, evaluating (|-14|) or (|14|) will both result in 14, as the absolute value operation strips away the negative sign, providing you with the magnitude of the number. It's essential to remember that evaluating an absolute value will never yield a negative result—it's the concept of measuring 'how much,' without any direction implied.

Absolute Value Expression

An absolute value expression is like a mathematical notation that extracts the positive magnitude of a number or an algebraic term. It is represented in writing by enclosing the number or term between two vertical bars like so: (|...|). This notation sends a clear message: 'Take the positive version of whatever is inside, please!'

Understanding how to work with these expressions is crucial for various math topics, from solving equations to analyzing functions. Here's a tip: An absolute value expression essentially asks a question—how far is the number inside from zero on the number line, regardless of direction? For example, when dealing with the expression (|-6 - 8|), you're asking 'How far is -6 from 8, ignoring if it's left or right?' The solution neatly sidesteps direction and purely focuses on distance.

Distance on the Number Line

Visualizing numbers on a straight path—this is the essence of a number line. It's a helpful tool for understanding operations, especially when it comes to distance. For example, if you're tasked with finding the distance between -6 and 8, imagine plotting these points on a number line and then simply measuring the space between them.

The number line inherently captures the essence of absolute values. Every point on a number line has a specific distance from zero, the origin. This distance, regardless of whether the point lies to the left or to the right of zero, is represented by the absolute value of the number. It's why (|-14|) becomes 14—the number line 'tells' us that -14 is 14 units away from zero. In other words, the number line is a physical representation that complements the concept of absolute values, clearly showing the distance between any two points is the absolute value of their difference.