Problem 12 - Beginning and Intermediate Algebra

Question

Subtract. See Examples 1 through 5 $$ -8-4 $$

Step-by-Step Solution

Answer

-12

Step 1: Identify the Operation

The problem asks us to subtract two numbers: \(-8\) and \(4\).

Step 2: Understand Subtraction with Negative Numbers

Subtracting a positive number from a negative one can be thought of as adding its negative. Thus, \(-8 - 4\) can be rewritten as \(-8 + (-4)\).

Step 3: Perform Addition of Negative Numbers

Now that the problem is rewritten as an addition of two negative numbers, we simply add their absolute values and keep the negative sign: \[-8 + (-4) = -(8 + 4) = -12.\]

Step 4: Conclusion

The result of the subtraction \(-8 - 4\) is \(-12\).

Key Concepts

Addition of Negative Numbers

When dealing with negative numbers, especially in subtraction, it's vital to understand how to handle their addition. Negative numbers are numbers with a value less than zero. They are typically represented with a "-" sign in front of the number. When we add negative numbers, we are essentially increasing a negative balance. This is akin to owing more money.

Here's how you can think of it:

• Convert subtraction into addition by adding the negative equivalent.
• For example, turning \(-8 - 4\) into \(-8 + (-4)\).
• Then proceed to add these negative numbers together.

To complete this, simply add the absolute values of the numbers (ignore the signs) and then apply the negative sign to your answer. This gives you the correct result when adding two negative numbers.

Negative Sign

The negative sign can often cause confusion, but once you understand its function, many math problems become much easier to solve. A negative sign indicates the opposite direction in terms of magnitude compared to positive numbers. It is the key indicator in moving values below zero.
When adding or subtracting negative numbers, it's essential to focus on the signs:

• If you see two negative signs in a row, like in \(-8 - 4\), the subtraction can be reinterpreted as adding negative numbers, e.g., \(-8 + (-4)\).
• The negative sign can also tell us about the result; combining negative numbers keeps the result negative.

Understanding the role of the negative sign aids in preventing mistakes and simplifying calculations with negative numbers.

Absolute Values

Absolute values play a crucial role when working with negative numbers. The absolute value of a number is its distance from zero on a number line, regardless of direction. It's essentially the non-negative "size" of the number.
When you are adding negative numbers, this concept becomes particularly helpful because:

• You first consider the absolute values to perform the arithmetic operation.
• In the example of \(-8 + (-4)\), you take the absolute values \(8\) and \(4\) and add them together.
• After calculating the sum of the absolute values, always reapply the negative sign to the result, thus representing the original direction.

Understanding absolute values ensures the calculations are consistent, especially which simplifies when negative numbers are in play.