Problem 18 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ 3+(-6) $$

Step-by-Step Solution

Answer

3 + (-6) = -3.

Step 1: Identify the Terms

The expression given is \(3 + (-6)\). Here you have two numbers to add: 3 and -6.

Step 2: Understand the Operation

Adding a negative number is the same as subtracting its positive counterpart. So, \(3 + (-6)\) is the same as \(3 - 6\).

Step 3: Perform the Subtraction

Subtract 6 from 3. This can be done by thinking of it on the number line, where you move 6 units to the left from 3, resulting in \(3 - 6 = -3\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are typically used to represent values below a baseline, such as temperatures below zero degrees Celsius or depths below sea level.
They can also be found in financial contexts to show a deficit or debt.
Negative numbers are represented with a minus sign (-) before the number. For example, -1, -2, and -3 are all negative numbers.

• Zero is neither a negative nor a positive number; it stands on its own.
• Negative numbers become more negative as their value decreases.
• When adding negative numbers to positive numbers, the negative sign tells us to subtract the absolute value of the negative number from the positive number. For example, in the exercise '3 + (-6)', the negative sign before 6 indicates a move in the negative direction.

Using the Number Line

A number line is a visual tool that helps to understand operations with numbers, including addition and subtraction.
It is a straight line with numbers placed at equal intervals along its length, extending infinitely in both positive and negative directions.

• Positive numbers are on the right side of zero, while negative numbers are on the left.
• To add a positive number, move to the right on the number line.
• To add a negative number, move to the left, which is the same as subtracting.

Using the number line for the exercise '3 + (-6)', you start at 3 and move 6 units to the left.
This method helps visualize the result as -3.
It's a powerful way to grasp the effect of negative numbers in an addition problem.

Subtraction as Addition of Negatives

Subtraction can sometimes feel confusing, but it can be simplified by understanding subtraction as the addition of negative numbers.
This means that when subtracting, you can think of it as adding a number with a negative sign.

• This is especially useful in cases where one has to subtract a larger number from a smaller one.
• For example, the expression '3 - 6' can also be written as '3 + (-6)'.

Both approaches yield the same result, as shown in our exercise where '3 - 6' and '3 + (-6)' both result in -3.
Viewing subtraction in this way simplifies calculations and provides a more cohesive understanding of the relationship between addition and subtraction.