Problem 88 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. $$ |-4-2|-|-8-1| $$

Step-by-Step Solution

Answer

-3

Step 1: Evaluate the first absolute value expression

First, calculate the value inside the absolute value on the left: \(-4 - 2 = -6\). Now take the absolute value: \(|-6| = 6\).

Step 2: Evaluate the second absolute value expression

Next, calculate the value inside the absolute value on the right: \(-8 - 1 = -9\). Now take the absolute value: \(|-9| = 9\).

Step 3: Subtract the absolute values

Finally, subtract the two absolute values obtained in the previous steps: \(6 - 9 = -3\).

Key Concepts

absolute value

Absolute value is a fundamental concept in mathematics, particularly in algebra. The absolute value of a number is its distance from zero on the number line, disregarding any negative sign. In essence, it turns any number into a non-negative number.
To express the absolute value, we use vertical bars, like this: \(|x|\).
For example:

• \(|2| = 2\)
• \(|-3| = 3\)

Absolute values are always non-negative and play a crucial role in many algebraic operations. In the given exercise, we evaluate the absolute values of \(-4 - 2\) and \(-8 - 1\) separately.

subtraction in algebra

Subtraction is one of the four basic arithmetic operations and involves taking one quantity away from another. In algebra, it is essential to follow the order of operations accurately. The operation follows the rule:
\text{Result} = \text{First Number} - \text{Second Number}
When dealing with negative numbers, it's crucial to pay attention to the signs. Negative signs can significantly affect the result.
For example:

• \(-4 - 2 = -6\)
• \(-8 - 1 = -9\)

In the example, we first deal with the expressions inside the absolute value bars by performing the subtraction.

negative numbers

Negative numbers are numbers less than zero, represented with a minus sign \((-\)). Understanding how to handle negative numbers is important in algebra. They often come up in subtraction and within absolute value operations.
When subtracting with negative numbers, follow these guidelines:

• Subtracting a positive number from a negative number makes it more negative. E.g., \(-4 - 2 = -6\).
• Subtracting a negative number from another negative number involves adding their absolute values together.

In the exercise, you handle the subtraction inside the absolute value first, resulting in negative outcomes. Once you have completed the subtraction, applying the absolute value converts the results into positive numbers before the final operation.