Problem 27 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ 6.3+(-8.4) $$

Step-by-Step Solution

Answer

The result is -2.1.

Step 1: Identify the Numbers

We have two numbers to add: 6.3 and -8.4. The positive number is 6.3, and the negative number is -8.4.

Step 2: Determine the Operation

Since we are dealing with a positive and a negative number, we need to find the difference between their absolute values. The operation will be subtraction.

Step 3: Subtract the Absolute Values

Calculate the difference between the absolute values of 8.4 and 6.3. \( 8.4 - 6.3 = 2.1 \).

Step 4: Determine the Sign of the Result

Since the larger absolute value is from the negative number (-8.4), the result will take the negative sign. Thus, the result is -2.1.

Key Concepts

Understanding Absolute Value

The concept of absolute value is foundational when working with integers, especially when adding and subtracting them. Absolute value refers to the distance of a number from zero on the number line, without considering the direction. It is denoted by vertical bars on either side of the number. For example, the absolute value of -8.4 is written as \(|-8.4|\), which equals 8.4. Similarly, the absolute value of 6.3 is \(|6.3|\), which is naturally 6.3.

Absolute values help determine how to handle operations between positive and negative numbers. When adding -8.4 and 6.3, we are essentially finding the difference between their absolute values. If a problem requires you to subtract numbers with different signs, the absolute value can guide which operation should be used.

Dealing with Negative Numbers

Negative numbers are simply numbers that are less than zero. They have a minus sign (-) in front of them and are located to the left of zero on the number line. For example, in the exercise, we have a negative number, -8.4. Negative numbers can be a bit tricky because they require us to think differently when performing mathematical operations.

When you add a negative number, it is like subtracting its positive counterpart. For instance, adding -8.4 to 6.3 means you actually take away 8.4 from 6.3. Visualizing this on a number line can be very helpful. Imagine starting at 6.3 on the number line and moving 8.4 units to the left. The position you land on is the result of the operation with the negative number. Such visualization simplifies understandings, like why adding -8.4 results in a smaller number than you start with.

Grasping the Subtraction Operation

Subtraction is one of the basic arithmetic operations, but it becomes more interesting when working with integers and their signs. In this context, subtraction involves finding the difference between the absolute values when dealing with numbers of different signs.

To subtract, determine which of the numbers has the larger absolute value, regardless of its sign. Then, subtract the smaller absolute value from the larger. In our example, the larger absolute value is 8.4 (from -8.4), and you subtract 6.3 (from 6.3) from it, resulting in 2.1. However, because -8.4 has the larger absolute value, the result takes on the negative sign, which gives us -2.1.

Thus, the sign of the difference is determined by the number with the larger absolute value. Understanding how subtraction interacts with negative numbers is crucial to solving such exercises efficiently.