Problem 19 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(12+(-12)\)

Step-by-Step Solution

Answer

0

Step 1: Identify the Numbers

Recognize that you need to add 12 and -12 together.

Step 2: Understand Opposite Numbers

Notice that 12 and -12 are opposites. When you add a number to its opposite, the result is 0.

Step 3: Calculate the Sum

Since 12 and -12 are opposites, their sum is 0: 12 + (-12) = 0.

Step 4: Verify with Number Line

As a check, if you start at 12 on the number line and move 12 units to the left (because of the -12), you return to 0.

Key Concepts

Opposite Numbers

Opposite numbers are pairs of numbers that are the same distance away from zero on a number line, but in opposite directions. For example, 12 and -12 are opposite numbers because they are both 12 units from zero—one to the right and one to the left.

When you add an integer and its opposite, the result is always zero. This is because they cancel each other out.

Consider the pairs:

• 5 and -5
• 7 and -7
• 20 and -20

In each case, adding the numbers together gives you 0. Understanding opposite numbers is essential for learning how to add integers effectively.

Number Line

A number line is a straight line with numbers placed at equal intervals along its length. Numbers to the right of zero are positive, and numbers to the left are negative. The number line is a helpful visual tool for understanding addition and subtraction of integers.

To use a number line for adding integers:

• Start from the first number.
• Move right for positive numbers.
• Move left for negative numbers.

For example, if you start at 12 and need to add -12, you move 12 units to the left. You will end up at 0. This confirms that 12 + (-12) = 0.

A number line is especially useful when dealing with more complex additions and subtractions, giving you a clear visual representation of the process.

Sum of Numbers

The sum of two numbers is the result you get when you combine them through addition.

For example:

• 10 + 5 = 15
• -3 + 7 = 4
• 6 + (-6) = 0

When adding integers, if one number is positive and the other is negative, think of the negative number as a debt and the positive number as an asset. If the debt and the asset are equal, they cancel each other out, resulting in a sum of zero.

Understanding the concept of adding positive and negative integers helps simplify complex problems and builds a foundation for advanced math concepts.