Problem 46 - Calculus with Concepts in Calculus

Question

Evaluate the expression. $$ |-5|-|5| $$

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Thus, the absolute value of any number is always non-negative. In this step, we will evaluate the absolute values separately for the numbers given.

Step 2: Evaluate \(|-5|\)

The absolute value of \(-5\) is the distance from zero, which is simply 5 because the absolute value strips away the negative sign. Thus, \(|-5| = 5\).

Step 3: Evaluate \(|5|\)

Similarly, the absolute value of \(5\) is just 5, since it is already a positive number. Therefore, \(|5| = 5\).

Step 4: Subtract the Absolute Values

Now that we have \(|-5| = 5\) and \(|5| = 5\), we can substitute these into the original expression: \(5 - 5\).

Step 5: Simplify the Expression

Perform the subtraction: \(5 - 5 = 0\).

Key Concepts

Distance from Zero

The concept of distance from zero is a critical part of understanding absolute values. Imagine a number line where all numbers are laid out in a straight line, with zero at the center. The distance from zero refers to how far a number is from the point designated as zero, without considering which direction you travel on the number line.
Every number has a distance from zero. For example:

• The number 5 is five units away from zero.
• The number -5 is also five units away from zero, just in the opposite direction.

This is why the absolute value, which measures this distance, is always a non-negative number. It tells us how far a number is from zero, not which side of zero it's on.

Evaluate Absolute Value

Evaluating absolute values involves finding the non-negative distance of a given number from zero, disregarding whether the number is positive or negative. This means that when you see the notation \(|x|\), you are simply being asked to determine how far x is from zero on the number line.
Let's break it down with a couple of examples:

• For \(|-5|\), we remove the negative sign to get 5, because -5 is 5 units from zero.
• For \(|5|\), the number is already positive, and its distance from zero is still 5.

Thus, evaluating absolute values is essentially an exercise in measuring these distances, which simplifies problem-solving and arithmetic tasks involving absolute value expressions.

Number Line

The number line is a fundamental tool in mathematics that helps us visualize numbers and their positions relative to each other. It extends infinitely in both directions with zero at its center. Numbers on the right of zero are positive, and those on the left are negative.
The absolute value concept uses the number line to describe the magnitude of a number's distance from zero. On this line:

• Each step or unit distance represents an integer value.
• The further right a number is, the larger its value.
• Every movement to the left moves you into negative territory, but each number's distance from zero remains a key metric.

By using a number line, you can easily evaluate absolute values and understand properties of numbers, especially when determining their absolute values as purely distances, irrespective of their direction.