Problem 67 - College Algebra Essentials

Question

Express the distance between the given numbers using absolute value. Then fi nd the distance by evaluating the absolute value expression. 2 and 17.

Step-by-Step Solution

Answer

The distance between 2 and 17 is 15.

Step 1: Understanding concept of absolute value

The absolute value of a number is its distance from 0 on the number line. It's denoted as \( |a| \). So, for any number \( a \), \( |a| = a \) when \( a \geq 0 \) and \( |a| = -a \) when \( a < 0 \). Now, we use this concept to find the distance between the two given numbers.

Step 2: Subtract the numbers

We subtract the two given numbers, either 17 - 2 or 2 - 17. We can subtract in any order because we're going to find absolute value of the difference, which will always be positive.

Step 3: Evaluate the absolute value

Take the absolute value of the difference. This gives the distance between the numbers. If we subtract 17 - 2, we get 15. The absolute value of 15 is 15, which is positive, so the distance is 15. If we subtract 2 - 17, we get -15. The absolute value of -15 is 15, which is also positive, so the distance is 15.

Key Concepts

Understanding the Number Line

When learning mathematics, one of the fundamental tools you will use is the number line. Imagine a horizontal line where each point represents a number. At the center of this line is 0, the origin, and from there, numbers increase to the right and decrease to the left. This visualization helps you see numbers not just as abstract entities, but as positions in space.

Using a number line, you can easily compare numbers to see which is greater or find their position relative to each other. It's like a map of numbers - each has its own unique location. Numbers to the right of 0 are positive, while those to the left are negative, and the further away from 0 they are, the greater their magnitude.

Grasping the Distance Between Numbers

The distance between numbers on the number line is a measure of how far apart they are. This distance is always a positive value, irrespective of the direction. To find the distance between two numbers, you subtract one from the other and disregard any negative sign that might result. This is comparable to finding out how many steps it would take to walk from one number to the other.

For example, the distance between the numbers 2 and 17 can be found by calculating either 17 - 2 or 2 - 17. Even though 2 - 17 gives a negative result, the actual distance we're interested in - the number of steps from 2 to 17 - is positive. We're simply looking for the magnitude of that difference.

Evaluating Absolute Value

To evaluate the absolute value means to determine the non-negative value of a number. The absolute value symbol (| |) denotes this operation. It translates the question 'how far is this number from zero?' into a numerical answer.

When you're given a difference like 17 - 2 or 2 - 17, evaluating the absolute value |17 - 2| or |2 - 17| will both yield the positive number 15. This positive number represents the distance on the number line, regardless of direction. So, the concept of absolute value is crucial in ensuring that distance is always expressed as a non-negative figure, aligning with the real-world notion that distance cannot be negative.