Problem 30 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$-96+(-31)$$

Step-by-Step Solution

Answer

The result of the addition -96 + (-31) is -127.

Step 1: Understand the Problem

We need to combine the numbers \(-96\) and \(-31\) using the rule for addition of positive and negative numbers. Since both numbers are negative, we will add their absolute values and keep the result negative.

Step 2: Find Absolute Values

Determine the absolute values of the numbers involved. The absolute value of \(-96\) is \(96\), and the absolute value of \(-31\) is \(31\).

Step 3: Add Absolute Values

Add the absolute values we found: \(96 + 31 = 127\).

Step 4: Apply the Negative Sign

Since we are adding two negative numbers, the resulting sum will also be negative. Therefore, \(-96 + (-31) = -127\).

Key Concepts

Understanding Absolute Value

Absolute value is a concept that expresses the magnitude of a number without considering its sign. Think of it as the distance a number is from zero on the number line. For any number, whether positive or negative, the absolute value is always non-negative. For example, the absolute value of

• \(-96\) is \(96\)
• \(-31\) is \(31\)
• \(+50\) is also \(50\)

It's like measuring how far apart two points are, regardless of direction. When you see an absolute value symbol \(|x|\), it just means to disregard whether \(x\) is positive or negative and focus on its actual size.
Remember, the absolute value gives us a standard way to compare or add numbers, which is essential when combining them, especially when they have different signs.

Absolute values, by ignoring the signs, help us easily perform arithmetic operations like addition and subtraction by simplifying the magnitude of each number.

Navigating Negative Numbers

Negative numbers represent values less than zero. They are usually marked with a minus sign (\(-\)). Understanding their behavior in arithmetic operations is crucial.
For negative numbers:

• When you add a negative number, you're essentially moving left on the number line.
• Adding \(-96\) means you start from where you are and move 96 units left.
• Negative numbers can often seem daunting, but their logic is straightforward: less than zero means a deficit or reduction.

When dealing with negative numbers:

• Combine negative numbers by adding their absolute values, then append a negative sign to the result.
• For example, in \(-96 + (-31)\), first compute \(|-96| + |-31| = 96 + 31 = 127\).
• Only after computing the sum do you reapply the negative sign to get a final result: \(-127\).

This method tells us how negative numbers interact with each other and helps in handling operations efficiently.

Rules for Adding Integers

The addition of integers follows certain rules, particularly when dealing with numbers that have different signs. These rules allow us to predict and accurately calculate the result.

• **Adding positive integers:** Simply add the numbers as you usually would, as their sums will always be positive.
• **Adding negative integers:** Add their absolute values, and apply a negative sign to the result. For instance, \(-96 + (-31) = -127\).
• **Adding a positive and a negative integer:** Subtract the smaller absolute value from the larger absolute value, and take the sign of the number with the larger absolute value.

It’s crucial to remember these rules:

• If the signs are the same, add the absolute values.
• If the signs are different, subtract the smaller absolute value from the larger one, maintaining the sign of the larger.

By following these integer addition rules, you can handle any kind of integer addition smoothly, whether it involves just positive numbers, just negative numbers, or a mix of the two. These rules help provide consistency and accuracy in computations.