Problem 28 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$9+(-1)$$

Step-by-Step Solution

Answer

The result of the expression \(9 + (-1)\) is 8.

Step 1: Understanding the Problem

We are asked to solve the expression \(9 + (-1)\). Here, we need to combine a positive number and a negative number.

Step 2: Apply the Rule of Addition

The rule for adding a positive number and a negative number is to subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value.

Step 3: Calculate the Absolute Values

Compute the absolute values of the numbers: \(|9| = 9\) and \(|-1| = 1\).

Step 4: Perform the Subtraction

Subtract the smaller absolute value from the larger: \(9 - 1 = 8\).

Step 5: Determine the Sign

Since 9 has the larger absolute value and is positive, the result will also be positive.

Step 6: Write the Final Answer

Combine the result and the sign from Step 4 and Step 5 to find the answer: \(9 + (-1) = 8\).

Key Concepts

Understanding Absolute Value

Absolute value is a crucial concept when dealing with integers, especially when you are adding or subtracting them. The absolute value of a number is simply how far it is from zero on the number line, disregarding its sign. This means that both positive and negative numbers have a distance, or magnitude, which is always non-negative.

Here's how we determine absolute value:

• The absolute value of a positive number is just the number itself, since it is already a distance from zero. For example, \(|9| = 9\).
• The absolute value of a negative number is its opposite positive number. For instance, \(|-1| = 1\).

Remember, in mathematical terms, this can be summarized as \(|x|\) if \(x\) is positive or zero, and \(-x\) if \(x\) is negative.

In our exercise, recognizing the absolute values allows us to correctly combine numbers of different signs. This approach simplifies calculations and ensures accurate results.

Positive and Negative Numbers

Understanding positive and negative numbers is essential when adding or subtracting integers. Positive numbers represent values greater than zero, while negative numbers represent values less than zero. In the realm of integers:

• Positive numbers are often written without a sign, e.g., 9, and lie to the right of zero on the number line.
• Negative numbers are always accompanied by a negative sign, e.g., -1, and lie to the left of zero on the number line.

When you add integers, especially a combination of positive and negative numbers, the result depends on their absolute values and their direction (positive or negative).

In our example, we combined a positive number (9) with a negative number (-1). Each number's direction influenced the final sum, with the larger absolute value (9) determining the final sign (positive). Hence, the result of adding 9 and -1 is 8, which remains positive because 9 is greater than 1 in absolute terms.

Integer Subtraction Simplified

Integer subtraction can often be seen as the addition of a negative. The expression \(9 + (-1)\) is essentially the same as \(9 - 1\). This is because subtracting a number is equivalent to adding its negative counterpart.

To solve such problems, here’s a simple approach:

• Convert the subtraction operation into adding a negative if helpful (e.g., \(9 - 1\) becomes \(9 + (-1)\)).
• Find the absolute values of both numbers involved.
• Subtract the smaller absolute value from the larger.
• Assign the sign of the number with the larger absolute value to the result.

This method ensures a seamless transition between subtraction and addition tasks, making it easier to compute with clarity. That’s why in our exercise, we found that \(9 + (-1)\) equals 8, by subtracting the absolute values and keeping the sign of the number with the larger absolute value.