Problem 64 - Algebra 1: Concepts and Skills

Question

Find the sum. $$ 6.5+(-3.4) $$

Step-by-Step Solution

Answer

The result is +3.1

Step 1: Take the absolute values

First, it is necessary to take the absolute values of the numbers. The absolute value of a number is its value regardless of sign. In this case, the absolute values will be 6.5 and 3.4.

Step 2: Perform subtraction

Now, subtraction is performed between the absolute values. The result would be \(6.5 - 3.4 = 3.1\)

Step 3: Assign the sign of the larger number

After the subtraction, the sign of the original number with the larger absolute value is assigned. In this case, 6.5 is larger than 3.4 and its original sign is positive. Therefore, the result is positive, which means the answer is +3.1.

Key Concepts

Absolute Value

When we speak about the absolute value of a number, we're referring to the distance of that number from zero on a number line, regardless of its direction. Think of it as the 'numerical value' without considering the sign. For instance, while working with integers or decimals such as 6.5 or -3.4, knowing their absolute values is essential.

The symbol for absolute value is two vertical bars flanking the number: for example, the absolute value of -3.4 is denoted as \( |{-3.4}| = 3.4 \). This concept is particularly useful when you need to find the magnitude of a number during operations such as addition and subtraction, as it simplifies the problem into one that deals with non-negative numbers only.

Subtraction of Integers

The subtraction of integers can sometimes confuse students, especially when dealing with negative numbers. However, if you break it down into a series of steps, it becomes easier to understand. First, find the absolute value of each integer, as explained in the previous section. Then, subtract these absolute values like you would any positive numbers.

The tricky part is determining the final sign of the answer. The rule of thumb is that the result takes the sign of the number with the larger absolute value, which is the farther number from zero on the number line. Therefore, when subtracting 3.4 from 6.5, even though 3.4 is negative, the outcome is positive because 6.5 has a larger absolute value.

Arithmetic Operations

Arithmetic operations are the foundation of all mathematics and they include addition, subtraction, multiplication, and division. In the context of our example with 6.5 and -3.4, we're combining the operations of addition and subtraction.

When adding a positive number and a negative number, the process is more akin to subtraction. We subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value. This mirrors our natural number sense – if you move forward 6.5 steps and then back 3.4 steps, you're ultimately 3.1 steps ahead.

Understanding the Rules

It's crucial to understand that arithmetic operations with integers follow certain rules and patterns. Mastery of these allows for solving complex equations and understanding fundamental mathematical concepts. Remember, practice with a variety of problems helps solidify these operations in your skill set.