Evaluate: $|8|-|-6|$
The expression we need to evaluate is \(|8| - |-6|\). The absolute value of a real number \(x\), denoted as \(|x|\), is the distance of the number from zero on the number line. Specifically, for any real number \(x\), its absolute value is defined as follows:
This means that the absolute value of a number is always non-negative. We will use this definition to evaluate each term in our expression separately.
First, we evaluate \(|8|\). Since \(8 > 0\), the definition of the absolute value tells us that \(|8| = 8\).
Next, we evaluate \(|-6|\). Since \(-6 < 0\), the definition of the absolute value tells us that \(|-6| = -(-6) = 6\).
Now, we substitute these evaluated values back into the original expression:
\[|8| - |-6| = 8 - 6\]
Performing the final subtraction, we get \(8 - 6 = 2\).
By applying the definition of absolute value to each term, we found that \(|8| = 8\) and \(|-6| = 6\). Subtracting the second from the first gives us \(8 - 6 = 2\). Thus, the value of the expression is \(\boxed{2}\).
Absolute value is essential in measuring distances, where direction doesn't matter. For instance, if you are walking along a street, your distance from your house is always non-negative whether you walk in a positive or negative direction on the street's coordinate system. Similarly, if you have \$8 in your wallet and you owe someone \$6, you can represent your net balance using absolute value concepts to understand the magnitude of your financial position. Understanding absolute values allows you to calculate differences in values regardless of whether the initial numbers are positive or negative.