Problem 67 - Algebra and Trigonometry

Question

Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. $$2\quad and\quad 17$$

Step-by-Step Solution

Answer

The distance between 2 and 17 is 15.

Step 1: Understand the Concept of Absolute Value

The concept of absolute value, symbolized by | |, is a way to express non-negative distance or difference. It removes any negative sign from a number, making it positive. It's best visualized on a number line, where it represents distance from zero.

Step 2: Apply the Concept to the Given Numbers

We're given the numbers 2 and 17. The distance between them is given by the absolute value of the difference between them. So, the expression that represents this distance is \( |2 - 17| \) or \( |17 - 2| \). Note that it doesn't matter which number is subtracted from which, because absolute value will make the result positive.

Step 3: Evaluate the Absolute Value Expression

To find the distance between the numbers, evaluate the absolute value expression. Either \( |-15| \) or \( |15| \) will yield the same result, which is 15.

Key Concepts

Distance Between Numbers

When we talk about the distance between two numbers, we're often interested in knowing how far apart they are on the number line. This concept is significant because the distance can help us compare numbers or understand their difference, irrespective of their order or sign.

There are a few points to keep in mind about finding distances between numbers:

• The distance is always non-negative, which means it's either positive or zero.
• We use absolute value to ensure this positivity.
• Calculating the distance involves subtracting one number from the other and then taking the absolute value of the result.

For example, if you have numbers like 2 and 17, the distance calculation would involve finding either \( |2 - 17| \) or \( |17 - 2| \). Both expressions reflect the notion that distance is absolute—it doesn't matter whether we subtract 17 from 2 or vice versa—because absolute value will always ensure the outcome is positive.

Evaluating Expressions

Evaluating expressions, especially those involving absolute values, follows a straightforward process. Once you have your numbers from which to calculate the difference, the next step is simple arithmetic, followed by finding that absolute value.

Here's how you can evaluate an expression like \( |2 - 17| \):

• First, perform the subtraction inside the absolute value: 2 - 17, which equals -15.
• Next, apply the absolute value to \( -15 \). The absolute value of a negative number is its positive counterpart, so \( |-15| = 15 \).

The result of these steps gives us the distance: 15. Evaluating expressions using absolute value is powerful in helping us determine how far two numbers are apart on a number line, safely removing any sign concerns.

Number Line Visualization

Visualizing numbers on a number line is a fantastic way to understand abstract concepts in a concrete form. Think of the number line as a straight path with evenly spaced tick marks, each representing a unique number.

Here's how number lines can help:

• They offer a visual reference that helps conceptualize the size of numbers and the space or distance between them.
• By plotting points on this line, you can directly see the gap or interval between two numbers like 2 and 17.
• The visualization makes it easier to grasp that \( |2 - 17| = 15 \) simply reflects the number of steps between 2 and 17.

Crucially, a number line shows us that distance is always positive, as it indicates the magnitude of separation without regard to direction. Whether counting forward or backward, the line reflects the true distance, just like absolute value captures in numerical form.