Problem 68 - Elementary and Intermediate Algebra

Question

Find each absolute value. $$ |-25.3| $$

Step-by-Step Solution

Answer

The absolute value is 25.3.

Step 1: Understand the Absolute Value

The absolute value of a number represents the distance of the number from zero on the number line, regardless of direction. Thus, absolute value is always non-negative.

Step 2: Remove the Negative Sign

Since absolute value is the distance from zero, the negative sign in front of a number becomes irrelevant. Therefore, the absolute value of |-25.3| is simply the positive version of 25.3.

Step 3: Write the Answer

Based on the definition of absolute value, |-25.3| = 25.3. The positive value, 25.3, is the final answer.

Key Concepts

Understanding the Number Line

The number line is an essential tool in mathematics that helps us to understand concepts like absolute value. Picture a straight, horizontal line marked with numbers at evenly spaced intervals. Zero is usually placed at the center, with positive numbers extending to the right and negative numbers extending to the left.

Here's how you might visualize it:

• Large numbers, whether positive or negative, are found further away from zero.
• Every number has a point on this line that represents its exact position.
• Each step to the right means adding one, while each step to the left means subtracting one.

The number line helps to show that moving towards the positive side or negative side increases the distance from zero, which is crucial for understanding absolute value.

Defining a Positive Number

Positive numbers are numbers greater than zero. They are found on the right side of the number line and typically have no sign in front of them, although a "+" can be used to specify positivity explicitly. Here's what makes them interesting:

• Positive numbers are used in everyday life to represent quantities that increase, such as money, height, or temperature above zero.
• They are the opposite of negative numbers, which are found to the left of zero and indicate a decrease or shortage.

In the context of absolute value, understanding positive numbers helps because when evaluating the absolute value, you always end up with a non-negative result, making negative numbers positive in the process.

Understanding Distance from Zero

Distance from zero is a straightforward concept but is very important in understanding absolute value. On the number line, distance measures how far a number is from the central point, zero.

Key points about this distance include:

• Distance is always non-negative; it's like measuring the distance between two points in space, and doesn't depend on direction.
• The absolute value of a number is basically this distance from zero, emphasizing why the result is non-negative.
• For instance, both 25.3 and -25.3 have the same absolute value because they are both 25.3 units away from zero, despite being on opposite sides of the number line.

By focusing on the distance alone, rather than the direction, you get a straightforward way to evaluate absolute value.