Problem 16 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ 8+(-6) $$

Step-by-Step Solution

Answer

The sum is 2.

Step 1: Identify the Numbers

In the expression \(8 + (-6)\), we need to identify the numbers involved. The first number is 8, and the second number is \(-6\).

Step 2: Understand the Sign of the Numbers

The number 8 is positive, and \(-6\) is negative. This means we have a positive number being added to a negative number.

Step 3: Consider the Absolute Values

The absolute value of 8 is 8, and the absolute value of \(-6\) is 6. When adding a positive and a negative number, consider the absolute difference.

Step 4: Subtract Smaller Absolute Value from Larger

Subtract the smaller absolute value from the larger: \(8 - 6 = 2\).

Step 5: Determine the Sign of the Result

Since the absolute value of the positive number (8) is larger than that of the negative number (6), the result takes the sign of the positive number. Hence, the result is positive.

Step 6: Write the Final Answer

The final sum is 2.

Key Concepts

Absolute Value

When tackling math problems involving integers, understanding the concept of absolute value is crucial. Absolute value is essentially the "distance" a number is from zero on the number line, regardless of direction. This means absolute value is never negative. For example:

• The absolute value of 8 is written as \(|8| = 8\).
• The absolute value of \(-6\) is written as \(|-6| = 6\), since six units separate \(-6\) and 0 on the number line.

Absolute value helps us compare and work with positive and negative numbers more directly. By converting numbers into their absolute forms, we can focus on the size of the numbers for addition, subtraction, or other operations, which simplifies the process.

Positive and Negative Numbers

Positive and negative numbers represent values on either side of zero on the number line. Positive numbers, like 8, lie to the right of zero and have a regular numeric value. Negative numbers, such as \(-6\), are to the left of zero and have a number preceded by a minus sign (-).

When adding or subtracting these numbers, it's essential to pay attention to their signs:

• Positive with positive: just add the values.
• Negative with negative: add their absolute values and prepend a minus sign to the result.
• Positive with negative (or vice versa): subtract the smaller absolute value from the larger one and follow the sign of the number with the greater absolute value.

Understanding these rules helps in performing operations like the sum of \(8 + (-6)\) where you consider the signs and effectively subtract the numbers due to their different signs.

Subtraction of Integers

Subtraction of integers can sometimes confuse students because it overlaps with addition of positive and negative numbers. When subtracting one integer from another, you are essentially adding its opposite. For example, subtracting \(-6\) from 8 can be viewed as adding its opposite, making it \(8 + 6\).

Let's consider the general rules:

• To subtract a positive number, add its negative.
• To subtract a negative number, add its positive equivalent.

In the given problem of \(8 + (-6)\), think of it as \(8 - 6\), since adding a negative number is like subtracting a positive one. By following this method, we determine the result accurately and find that \(8 - 6 = 2\), confirming the correct operation through substitution of operations.