Problem 65 - Elementary and Intermediate Algebra

Question

Add. $$ -6+(-8) $$

Step-by-Step Solution

Answer

-14

Step 1: Understanding the Problem

The problem requires us to add two negative numbers: -6 and -8. When adding negative numbers, we are essentially adding the absolute values of these numbers and then placing a negative sign in front of the sum.

Step 2: Identify the Absolute Values

First, identify the absolute values of both numbers: |-6| is 6 and |-8| is 8. These are the positive values of the numbers without considering their signs.

Step 3: Add the Absolute Values

Now, add the absolute values: 6 + 8 = 14. This result is obtained by adding the positive versions of the two numbers.

Step 4: Apply the Negative Sign

Since we are adding two negative numbers, the result must also be negative. Thus, we take the sum from the previous step (14) and apply a negative sign to get -14.

Key Concepts

Absolute Values

When dealing with negative numbers, it is essential to understand the concept of absolute values. An absolute value represents the magnitude of a number regardless of its sign. In simpler terms, it is the distance a number is from zero on the number line without considering whether it is to the left or the right.

• For positive numbers, the absolute value is the number itself. For example, the absolute value of 5 is 5: \(|5| = 5\).
• For negative numbers, the absolute value is the number without its sign. For instance, the absolute value of -6 is 6: \(|-6| = 6\).

Knowing how to find and use absolute values makes working with negative numbers much easier, especially when performing operations such as addition.

Integer Addition

Integer addition encompasses adding both positive and negative whole numbers. This operation can sometimes be tricky, particularly when negative numbers are involved. Here’s a simple breakdown to help you:

• When adding two positive numbers, just add them as usual, like \(3 + 4 = 7\).
• When adding a positive and a negative number, subtract the smaller absolute value from the larger one, and take the sign of the number with the larger absolute value. For instance, \(7 + (-3) = 4\) since 7 has a larger absolute value than -3.
• When adding two negative numbers, add their absolute values together, then place a negative sign in front of the result. An example of this is \(-6 + (-8) = -14\).

Understanding these basic rules of integer addition allows for easier computations and solving more complex problems.

Negative Numbers

Negative numbers are numbers less than zero, typically indicated by a minus sign (−). These numbers are located to the left of zero on the number line. They are used to represent values below a defined zero point, such as temperatures below freezing or debts.Working with negative numbers involves understanding their behavior during mathematical operations:

• Adding negative numbers to positive numbers often results in decreasing the positive number or making it negative, depending on which is larger in absolute value.
• Subtracting negative numbers is the same as adding their positive counterparts. For example, \(-5 - (-3)\) is equivalent to \(-5 + 3\), resulting in \(-2\).
• When multiplying or dividing negative numbers, signs play a crucial role. Multiplying or dividing two numbers with the same sign yields a positive product or quotient, whereas two numbers with different signs produce a negative result. For example, \((-2) \times (-3) = 6\) and \((-2) \times 3 = -6\).

Mastering negative numbers is crucial for effective problem-solving and application in real-world contexts.