Problem 2 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ 9+(-12) $$

Step-by-Step Solution

Answer

-3

Step 1: Identify the Numbers

We have two numbers: 9 and -12. Notice that one number is positive (9) and the other is negative (-12).

Step 2: Subtract Their Absolute Values

The absolute value of 9 is 9, and the absolute value of -12 is 12. We need to find the difference: 12 - 9 = 3.

Step 3: Determine the Sign of the Result

Since -12 has the larger absolute value (12) compared to 9 (which has the absolute value of 9), the result will take the sign of -12. Therefore, the final result is -3.

Key Concepts

Understanding Absolute Value

Absolute value is an important concept in mathematics. It represents the non-negative distance of a number from zero on the number line. For any number, whether positive or negative, the absolute value is always positive.
To find the absolute value of a number, you simply consider its numerical value without regard to its sign.
For instance, the absolute value of 9 is 9, and the absolute value of -12 is 12.

• Absolute value of a positive number (e.g., 9) is the same number: |9| = 9.
• Absolute value of a negative number (e.g., -12) is its positive counterpart: |-12| = 12.

Understanding the concept of absolute value is crucial when dealing with negative numbers and performing operations like addition.

Exploring Negative Numbers

Negative numbers are those found to the left of zero on the number line. They are typically less than zero and written with a minus sign (e.g., -12).
These numbers can represent "loss" or "below zero" in real-world scenarios, such as temperatures below freezing or debt in finances.
Understanding negative numbers involves knowing how they compare with positive numbers:

• Negative numbers are smaller than zero.
• A negative number is less than a positive number. For example, -12 is less than 9.
• When dealing with negative numbers in calculations, remember their relative positions on the number line and how they interact with absolute values.

Addition with Negatives

Adding a positive number to a negative number—or vice versa—involves considering their absolute values. This process may appear tricky at first, but with practice, it becomes second nature.
To correctly add integers when negatives are involved, follow these steps:

• First, identify both numbers and recognize their sign. One is usually positive while the other is negative.
• Subtract the smaller absolute value from the larger absolute value. For the example given, subtract the absolute value of 9 from 12, which results in 3.
• The final step is determining the sign of the result. This is done by taking the sign of the number with the larger absolute value. In our example, since 12 is greater than 9, we use the negative sign from -12. Thus, the result of 9 + (-12) is -3.

Remember, the key is to focus on the absolute values and apply the appropriate sign afterwards.

Step-by-Step Solution Guide

Approaching addition involving negative numbers through a step-by-step guide can make it easier to comprehend. Let's break down the given example of 9 + (-12).
Step 1: Identify the Numbers
We have two numbers: 9 (positive) and -12 (negative). Recognizing their signs is crucial.
Step 2: Subtract Their Absolute Values
Determine the absolute values: |9| = 9 and |-12| = 12. Proceed to subtract the smaller absolute value from the larger one: 12 - 9 = 3.
Step 3: Determine the Sign of the Result
Since the absolute value of -12 (12) is greater than that of 9 (9), the result will be negative.
Conclusively, 9 + (-12) yields -3.
Utilizing a structured approach clarifies the process of working with integers and makes integer addition straightforward.