Problem 81 - Prealgebra

Question

The problems below review addition, subtraction, multiplication, and division of positive and negative numbers, as covered in this chapter. Perform the indicated operations. $$8+(-4)$$

Step-by-Step Solution

Answer

The result is 4.

Step 1: Understand the Operation

The expression given is an addition of two numbers: 8 and -4. This requires combining a positive number with a negative number.

Step 2: Simplify using Rules for Addition

When adding a positive and a negative number, subtract the absolute values and keep the sign of the larger absolute value. Here, subtract the absolute value of -4 (which is 4) from the absolute value of 8 (which is 8).

Step 3: Perform the Calculation

Subtract 4 from 8: \[ 8 - 4 = 4 \]

Step 4: Determine the Sign

Since 8's absolute value is larger and is positive, the result will take the positive sign. Therefore, the final result is 4.

Key Concepts

Addition of Integers

Adding integers, especially when dealing with positive and negative numbers, might seem challenging. But it's quite simple once you understand the basic rules. When you add two integers, you need to consider their signs:

• If both integers are positive or both are negative, simply add their absolute values.
• If the integers have different signs, subtract the smaller absolute value from the larger one.
• Keep the sign of the number with the larger absolute value.

In the example we have, 8 is positive and -4 is negative. So, you take the absolute value of both and subtract the smaller from the larger: \( 8 - 4 = 4 \). The final answer is positive because 8 is larger than 4.

Rules for Negative Numbers

Dealing with negative numbers in math can be tricky, but understanding the rules can make things easier. Here are the key points:

• When two negative numbers are added together, the result is negative.
• When a positive number and a negative number are added, subtract the smaller number and keep the sign of the larger number's absolute value.

Negative numbers can often represent a debt or decrease, which is why addition involving them often results in subtraction. In our example problem, combining 8 with -4 involved subtracting the two because they have different signs.

Absolute Value Operation

The absolute value of a number is its distance from zero, regardless of direction on the number line. It's always positive, making it a useful tool in integer operations. Here’s why:

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is its positive counterpart.
• Absolute values help in comparing and subtracting numbers with different signs.

For example, in our calculation of \(8 + (-4)\), we used the absolute values 8 and 4 to determine the operation: \(8 - 4 = 4\). Absolute values simplified the process by turning it into a simple subtraction problem. This makes calculations involving different signed numbers much clearer and easier to handle.