Problem 5 - Algebra 2 and Trigonometry

Question

Find the value of each given expression. \(|8-3|\)

Step-by-Step Solution

Answer

The value of \\( |8-3| \\) is 5.

Step 1: Understand the Absolute Value Function

The absolute value of a number is its distance from zero on a number line, regardless of direction. For any number \( x \), the absolute value is represented as \(|x|\) and is always non-negative.

Step 2: Evaluate the Expression inside the Absolute Value

First, evaluate the expression inside the absolute value: \( 8 - 3 \). Subtract 3 from 8 to get 5.

Step 3: Apply the Absolute Value

Since the result of \( 8 - 3 \) is 5, apply the absolute value to this result. Since 5 is already positive, \(|5| = 5\).

Key Concepts

Distance on a Number Line

When we talk about distance on a number line, we are simply referring to how far apart two numbers are, without considering the direction. Imagine a straight line where integers (whole numbers) are evenly spaced. The absolute value concept helps us determine this distance from zero to any number. Whenever you calculate the absolute value of a number, such as \(|x|\), you're finding how far that number is from zero.

For example, if you have the number 5, its absolute value is \(|5|\), which equals 5, because it is 5 units away from zero. Similarly, for -5, \(|-5|\) is also 5, reflecting its 5-unit distance on the opposite side of zero. Remember, distance is always a positive measurement, so absolute values are always non-negative. Whether you move left or right on the number line, the absolute value provides a clear, positive measure of distance.

Evaluating Expressions

Evaluating expressions is like solving a puzzle. It involves simplifying or calculating within an equation to find a result. In the context of absolute values, the process includes another step: simplifying the expression inside the absolute value before applying the absolute value itself.

Using the expression \(8 - 3\) as our example, start by simplifying within the parentheses. Subtract 3 from 8, resulting in 5. After simplifying, you apply the absolute value function. If the result inside the absolute value is positive, as in our example, the number remains unchanged. However, if it had been negative, the absolute value would still render it positive, because the focus is on distance, not direction.

This methodology of evaluating expressions can be applied to more complex operations but starts with these simple steps. Always remember: simplify first, apply absolute value second.

Non-negative Numbers

Non-negative numbers are numbers that are zero or positive. These include all positive integers as well as zero. The importance of non-negative numbers comes into play especially when dealing with absolute values.

For any number you place inside an absolute value, the result will always be a non-negative number. For example, \(|5|\) remains 5, and \(|-3|\) becomes 3. Absolute value signifies that regardless of whether a number starts as positive or negative, the outcome will measure a non-negative distance.

Understanding non-negative numbers is crucial in various math applications. This concept guarantees that regardless of whether an initial value is negative, once you apply the absolute value, it aligns with our mathematical understanding of distance: always non-negative. This property repeats in many areas of math and helps keep calculations and their interpretations consistent and clear.