Problem 61 - Beginning and Intermediate Algebra

Question

Find each absolute value. \(-|6-3|\)

Step-by-Step Solution

Answer

-3

Step 1: Simplify Inside the Absolute Value

First, subtract the numbers inside the absolute value: Subtraction: \(6 - 3 = 3\)

Step 2: Apply Absolute Value

Next, find the absolute value of the result: \(|3| = 3\)

Step 3: Apply the Negative Sign

Finally, apply the negative sign outside the absolute value: \(-|3| = -3\)

Key Concepts

absolute value

The absolute value of a number measures how far that number is from zero on a number line, regardless of direction.
In simpler terms, it's the non-negative version of any given number. For example, the absolute value of both 5 and -5 is 5 because they are both five units away from zero.
We denote absolute value using vertical bars, like this: \(|x|\). So, \(||3|| = 3\) and \(||-3|| = 3\). Absolute value essentially 'removes' any negative sign in front of a number, converting it to its positive counterpart.

subtraction inside absolute value

The core idea behind subtraction inside absolute value is to simplify the expression under the absolute value first.
In our example, we have \(6 - 3\). We need to perform this subtraction before applying the absolute value.
\((6 - 3 = 3)\).
Now, we replace the original inside of the absolute value with 3, resulting in \(|3|\).
Always make sure to simplify expressions inside the absolute value before anything else.

negative sign application

Once we have simplified the expression within the absolute value and found its absolute value, we can apply any additional operations.
Our example ends with a negative sign outside the absolute value: \(-|6-3|\).
After getting the absolute value, which is 3, the last step is to apply the negative sign: \(-|3| = -3\).
Remember, the absolute value operation always comes before applying any outside operations, like adding or subtracting a sign.