Problem 32 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ -6.7+(-7.6) $$

Step-by-Step Solution

Answer

\(-6.7 + (-7.6) = -14.3\)

Step 1: Identify the Numbers

First, clearly identify the numbers you need to add. The numbers given here are \(-6.7\) and \(-7.6\).

Step 2: Apply the Rule for Adding Negative Numbers

When adding two negative numbers, the result is also negative. This is because you're essentially adding the magnitudes (absolute values) together, while keeping the negative sign. This means the operation becomes \(-|6.7| - |7.6|\).

Step 3: Calculate the Sum of the Magnitudes

Calculate the absolute values and sum them. \(6.7 + 7.6 = 14.3\). This value represents the combined 'size' of both numbers, ignoring the negative sign for now.

Step 4: Apply the Negative Sign to the Sum

Since both of the original numbers were negative, the final result is also negative. Thus, you apply the negative sign to the sum of the magnitudes: \(-14.3\).

Key Concepts

Understanding Absolute Values

When dealing with numbers, especially negative ones, it's important to understand the concept of absolute values. The absolute value of a number is its distance from zero on the number line, regardless of its direction. This means that both positive and negative numbers share the same magnitude of distance from zero. For example:

• The absolute value of 6.7 is 6.7
• The absolute value of -6.7 is also 6.7

Absolute values help us focus on the size or magnitude of a number without considering its sign. Consequently, when performing operations like addition or subtraction with negative numbers, determining the absolute value becomes an important step.

The Role of the Negative Sign

The negative sign is more than just a symbol—it tells us the direction of the number on the number line. A positive number is to the right of zero, while a negative number is to the left. In mathematical operations, retaining the correct negative sign ensures that the problem reflects real-world scenarios accurately.
For example, in the exercise where we add -6.7 and -7.6, the numbers have negative signs. This means, regardless of the sum of their absolute values (which is 14.3), they must remain negative. Thus, the final answer is negative. Handling the negative sign accurately is key when carrying out arithmetic operations.

Performing Mathematical Operations

In mathematics, operations such as addition, subtraction, multiplication, and division follow specific rules, particularly when negative numbers are involved. Adding two negative numbers involves:

• Calculating the absolute values of both numbers.
• Adding those absolute values together.
• Applying a negative sign to the result, since you combined two numbers on the left side of zero.

Following this method helps in simplifying the process and achieving an accurate result. For example, adding -6.7 and -7.6 involves:

• Finding their absolute values: 6.7 and 7.6.
• Adding them: 6.7 + 7.6 = 14.3.
• Since both numbers are negative, the final result is -14.3.

Recognizing these operations is crucial for tackling complex math problems.

Visualizing Using the Number Line

The number line is a useful tool for understanding how numbers interact, especially when adding negative numbers. On a number line:

• Numbers are placed in increasing order from left to right.
• Zero is the central point, with positive numbers extending to the right and negative numbers to the left.

When adding two negative numbers, moving leftward on the number line represents adding more negative value. For instance, starting from -6.7 and adding -7.6 means traveling further left—an additional 7.6 units from -6.7.
Visualizing problems in this way can help students understand the direction and value of numbers, reinforcing the rules of adding with negatives and aiding overall comprehension of mathematical operations.