Problem 67 - Elementary and Intermediate Algebra

Question

Find each absolute value. $$ |-6.1| $$

Step-by-Step Solution

Answer

The absolute value of -6.1 is 6.1.

Step 1: Understand the Problem

We need to find the absolute value of the number -6.1. The absolute value refers to the distance of a number from zero on the number line, regardless of direction.

Step 2: Apply Absolute Value Definition

The absolute value of a number is always non-negative. For any negative number -x, the absolute value is given by changing the sign, i.e., |-x| = x. Therefore, |-6.1| becomes 6.1.

Step 3: Verify the Solution

Finally, ensure that the definition of absolute value is correctly applied. Since -6.1 is a negative number, its absolute value is 6.1, which is non-negative as expected.

Key Concepts

Negative Numbers

Negative numbers are values less than zero, often represented with a minus sign (e.g., -6.1). On a number line, they appear to the left of zero. Negative numbers are crucial in many mathematical operations and concepts, including absolute value.
Negative numbers signal opposite directions or deficits. For instance:

• Negative temperatures indicate colder than zero degrees.
• Negative bank balances mean an overdraft or debt.

Understanding negative numbers involves recognizing their behavior in arithmetic operations.

• Adding a negative is equivalent to subtraction: \(-3 + (-2) = -3 - 2 = -5\).
• Multiplying two negative numbers gives a positive result: \((-3) \times (-2) = 6\).

When dealing with absolute values, negative numbers become significant because the process involves converting them into their positive counterparts.

Number Line

A number line is a visual representation of numbers laid out evenly along a straight line. It helps in understanding the concept of number order and operations. The center of the number line is typically zero, separating positive and negative numbers. Positive numbers appear to the right of zero, whereas negative numbers are positioned to the left.
Consider points of interest on the number line:

• A number's position represents its distance from zero.
• Negative numbers always lie left of zero, highlighting their less-than-zero value.

When finding absolute values, the number line illustrates a number's distance from zero disregarding direction, effectively transforming a negative movement into a positive distance. For the problem of \(-6.1\), visually assessing the distance to zero clarifies why the absolute value results in 6.1.

Non-negative Value

The term "non-negative value" refers to values that are zero or greater, never dipping below zero. In mathematics, when we find the absolute value, we convert the number into a non-negative value, regardless of its initial sign. Understanding non-negative values is key in diverse mathematical applications and ensures clarity in mathematical problems.
Key characteristics include:

• Zero is considered non-negative since it is neither positive nor negative.
• Non-negative values offer a straightforward representation of magnitude without direction.

In the context of absolute value, transforming \(-6.1\) into \(6.1\) demonstrates making the value non-negative. This transformation reflects the distance from zero on the number line as a positive value, which aligns with the definition of absolute value.