Question

Find the difference. $$ |8|-12.6 $$

Short answer

The difference is \(-4.6\).

Step-by-step solution

Determine the absolute value of 8

The absolute value of a number is its distance from 0, so the absolute value of 8 is 8 since it's 8 units away from 0 on the number line.

Subtract the given decimal

Now, subtract 12.6 from 8. This gives -4.6.

Key concepts

Absolute Value

Understanding the absolute value of a number is crucial when working with integers. The absolute value is defined as the distance a number is from zero on a number line, regardless of direction. In other words, it's a measure of how far a number is from the origin and is always a non-negative value. For instance, the absolute value of both -8 and 8 is 8, as each is 8 units away from 0 on the number line. This is expressed mathematically as:\[ |8| = 8 \] and \[ |-8| = 8 \]. The bars surrounding the number represent the absolute value operation.

Subtracting Decimals

Subtracting decimals might seem intimidating at first, but with the right approach, it's quite straightforward. To begin, align the decimal points of the numbers involved. If one number has fewer decimal places, you can add zeros to the end until they match. For example, consider subtracting 12.6 from 8. Since 8 can be thought of as 8.0, it aligns with 12.6 like this:\[ \begin{align*} &8.0 ewline -&12.6 \end{align*} \] You then subtract each digit, borrowing from the next column if necessary. However, given that 8 is smaller than 12.6, the result will be a negative number illustrating you owe some amount after the subtraction:\[ 8.0 - 12.6 = -4.6 \].

Subtracting a larger decimal from a smaller one as in the exercise illustrates a key concept: The magnitude of the result reflects the difference between the two decimal values, and the sign indicates which number was larger to begin with.

Number Line

The number line is a fundamental tool in mathematics that illustrates numbers as points on a line. Positive numbers are placed to the right of zero, while negative numbers are to the left. The number line is useful for visualizing operations such as addition, subtraction, and understanding the concept of absolute value. When we consider the problem \(|8| - 12.6\), imagine first locating the point 8 (since its absolute value is 8) on the right side of the number line and then moving left 12.6 units, as subtraction indicates a movement to the left. This movement to the left will take you past zero to land at -4.6. This visual representation shows why the result of the subtraction is a negative number.