Problem 18 - Intermediate Algebra

Question

Perform the operations. See Example 1 . $$ 3.1+(-5.2) $$

Step-by-Step Solution

Answer

The result is -2.1.

Step 1: Understand the Problem

We need to perform the addition operation of two numbers: 3.1 and -5.2. This is a problem involving addition of a positive number and a negative number.

Step 2: Identify the Sign Rule for Addition

When adding a positive number to a negative number, we subtract the smaller absolute value from the larger one and take the sign of the larger number.

Step 3: Calculate the Absolute Values

The absolute value of 3.1 is 3.1, and the absolute value of -5.2 is 5.2. Compare the absolute values to understand which is larger.

Step 4: Subtract the Smaller Absolute Value from the Larger One

Subtract 3.1 from 5.2: \[5.2 - 3.1 = 2.1\]

Step 5: Determine the Sign of the Result

Since the larger absolute value is from -5.2, the result takes the sign of -5.2. Therefore, the result is -2.1.

Key Concepts

Absolute Value

When dealing with numbers, whether positive or negative, understanding the concept of absolute value is crucial. The absolute value of a number is its distance from zero on the number line, without considering its direction or sign. This means that both positive and negative numbers have positive absolute values.
For example,

• The absolute value of 3.1 is 3.1.
• The absolute value of -5.2 is 5.2.

Absolute values are helpful in situations involving the addition or subtraction of numbers, especially when combining numbers with different signs. By referring to their absolute values, you can easily determine which number has a larger magnitude.
Keep in mind that the concept of absolute value essentially removes the sign or direction of a number, allowing you to focus solely on its size.

Sign Rule

The sign rule is an important guideline to follow when adding or subtracting numbers with different signs. When you're faced with adding a positive number to a negative one, or vice versa, follow this simple rule;
subtract the smaller absolute value from the larger absolute value, and then take the sign of the number with the larger absolute value.
This method ensures that the result reflects the overall direction indicated by the larger magnitude. For instance:

• Consider adding 3.1 and -5.2.
• The absolute value of 3.1 is 3.1, while for -5.2, it is 5.2.
• Since 5.2 is larger, the result will have the sign of -5.2, i.e., negative.

Using the sign rule allows for a consistent approach, ensuring that the result accurately represents the combined effect of the numbers involved in the operation.

Addition Operation

Adding numbers, whether positive or negative, is a basic mathematical operation that involves combining values. When the numbers have different signs, the addition operation becomes a mix of addition and subtraction. Essentially, you are adding up the magnitudes but considering the sign of the larger number.
To perform an addition operation:

• Calculate the absolute values of each number.
• Subtract the smaller absolute value from the larger one.
• Assign the sign of the larger number to the final result.

As an example, when adding 3.1 with -5.2,
you start by recognizing that their absolute values are 3.1 and 5.2, respectively.
Then, subtract the smaller (3.1) from the larger (5.2) to get 2.1.
Finally, since 5.2 is larger and its original number was negative, the result is -2.1.
This approach helps break down the addition process into straightforward steps, making it easier to handle numbers of differing signs.