Problem 4 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ -6+(-14) $$

Step-by-Step Solution

Answer

The sum is -20.

Step 1: Understand the Problem

The problem requires you to add two negative numbers: -6 and -14.

Step 2: Apply the Rule of Adding Negative Numbers

When adding two negative numbers, remember that the result will also be negative, and you simply add their absolute values. Here, add the absolute values of 6 and 14.

Step 3: Calculate the Sum of the Absolute Values

Add the absolute values: 6 + 14 = 20.

Step 4: Apply the Negative Sign to the Result

Since both numbers being added were negative, the final result is also negative. Therefore, the sum is -20.

Key Concepts

Absolute Value

When dealing with numbers, especially integers, the concept of absolute value can be very beneficial. The absolute value of a number is its distance from zero on the number line, without considering direction. Hence, for any given number whether it is positive or negative, its absolute value will be positive.

For instance:

• The absolute value of 6 is 6, which is written as \(|6| = 6\).
• The absolute value of -6 is also 6, expressed as \(|-6| = 6\).

In short, the absolute value negates any negative sign present with the number, focusing solely on the numerical value. Understanding absolute value is essential in operations like addition and subtraction, especially when dealing with negative numbers.

Integer Addition

Integer addition involves combining whole numbers, which can include both positive and negative values. Integer addition follows simple rules that help maintain consistency and accuracy in calculations. There are two primary rules useful for integer addition:

• When adding two positive integers, simply add their values, and the result is positive. For example, adding 3 and 5 gives 8.
• When adding two negative integers, their absolute values are added first, and the result is negative. For example, adding -6 and -14 involves adding 6 and 14 to get 20, then applying the negative sign to get -20.

These rules make it easier to perform calculations quickly and accurately, ensuring that you can handle both everyday arithmetic and more complex mathematical problems efficiently.

Negative Numbers

Negative numbers can often seem tricky, but with the right understanding, they become much less daunting. Negative numbers are those less than zero, represented with a minus sign. On a number line, they appear to the left of zero. In real life, negative numbers can represent many things, such as debts or temperatures below freezing.

When performing arithmetic operations with negative numbers, remember:

• Adding two negative numbers results in a negative sum (as with the example \(-6 + (-14) = -20\)).
• Adding a positive number to a negative number subtracts, in effect reducing the "negative" quality of the sum.

Understanding these principles, negative numbers transform from intimidating to manageable components of math.

Basic Arithmetic Operations

Understanding basic arithmetic operations is crucial for any mathematical calculations and includes addition, subtraction, multiplication, and division. Each of these operations has specific rules, especially when applied to negative numbers. Here, we'll focus on addition.

• Addition: When you add numbers, you are essentially combining their values. For negative numbers, this involves combining their absolute values and maintaining a negative sign.
• Consistency: The consistent application of arithmetic rules simplifies mathematical computations, whether for simple or complex problems.

By grasping these operations, students can solve exercises efficiently and with confidence, including those involving negative numbers that initially seem more complex.