Problem 39 - Elementary Algebra

Question

Find the sums. \(8+(-15)\)

Step-by-Step Solution

Answer

Answer: The sum of the integers 8 and -15 is -7.

Step 1: Understand the problem

Identify that the problem involves the addition of a positive and a negative integer: \(8 + (-15)\).

Step 2: Adding the integers

Since we are adding a positive and a negative integer, we can rewrite the sum as the difference between two positive integers by subtracting their absolute values: \(|8 - 15|\).

Step 3: Compute the difference

Calculate the difference between the two positive integers: \(|8 - 15| = |-7|\).

Step 4: Determine the sign of the sum

We know that the larger absolute value is associated with the negative integer, so the sum will have a negative sign: \(-7\).

Step 5: Write down the final answer

The sum of \(8+(-15)\) is \(-7\).

Key Concepts

Addition of Positive and Negative Numbers

When you are tasked with adding a positive number and a negative number, the process involves a few simple steps.
1. **Identify the numbers**: Note which number is positive and which is negative. In our example, we are adding 8 (positive) to -15 (negative).
2. **Determine the operation**: Since one number is positive and the other is negative, you effectively perform a subtraction, where the number with the largest absolute value determines the sign of the answer.
For instance, the absolute value of -15 is 15, which is larger than 8. Thus, the sum will be negative.

• This step is pivotal because it dictates whether you'll have more negatives or positives left over.
• The result is computed as \(8 + (-15) = 8 - 15\).

This simple approach helps to manage integer additions involving mixed signs.

Absolute Value

Absolute value is a crucial concept in arithmetic, particularly when dealing with positive and negative numbers.
It provides a measure of a number's distance from zero on the number line, without regard to direction.
In more simple terms, it's like forgetting the negative sign for a moment and just considering how far the number is from zero.

The absolute value of a number is denoted by placing the number within vertical bars, such as \(|x|\).

• For positive numbers: \(|8| = 8\)
• For negative numbers: \(|-15| = 15\)

Understanding absolute values simplifies the process of comparing, adding, or subtracting integers.
It allows you to focus on the quantity, making arithmetic operations more manageable, especially when identifying which integer has the greater absolute value in problems involving both positive and negative numbers.

Arithmetic Operations

Arithmetic operations are the basic calculations you perform with numbers: addition, subtraction, multiplication, and division.
In the context of this exercise, the emphasis is on addition, specifically involving integers of different signs.

When adding integers with different signs, the calculation resembles that of subtraction, potentially causing confusion.

• Consider always the absolute values: The operation transforms into a subtraction \( |8 - 15|\).
• Assess the direction of the result: The sign of the total sum mirrors the sign of the largest absolute value involved (in our example, negative due to -15).

This understanding aids in resolving arithmetic operations efficiently and prevents errors by treating each operator with its properties.
Moreover, grasping these fundamental arithmetic principles is crucial for advancing in more complex math topics.