Problem 24 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(-18+18\)

Step-by-Step Solution

Answer

0

Step 1: Identify the Numbers

The equation to solve is \( -18 + 18 \). Notice that one number is negative and the other one is positive.

Step 2: Understand Opposites

Understand that \( -18 \) and \( 18 \) are opposites. When you add a number to its opposite, the sum is always zero.

Step 3: Perform the Addition

Since \( -18 + 18 \) are opposites, their sum is \( 0 \). Therefore, \( -18 + 18 = 0 \).

Step 4: Check Using Number Line (Optional)

To check, you could use a number line: start at -18 and move 18 steps to the right. You will land at 0.

Key Concepts

opposites in mathematics

In mathematics, opposites are pairs of numbers that, when added together, equal zero.
These numbers are often called additive inverses.
The basic idea is that for any number, there exists another number such that their sum is zero.
For example:

• -18 and 18
• -5 and 5
• -100 and 100

These pairs are opposites.
The relationship between opposites is fundamental in integer addition.
By recognizing that one number is the exact negative of another, you can easily solve addition problems.

sum of opposites

The sum of opposites always equals zero.
This concept is important in understanding how integers work together.
When you add a number to its opposite, no matter how large or small, the result is zero.
Consider the equation \(-18 + 18\):

• -18 is the opposite of 18
• By adding them, you perform the operation: \(-18 + 18 = 0\)

The result is zero because the negative value cancels out the positive value.
This principle helps simplify many arithmetic problems.

zero property of addition

The zero property of addition states that any number added to its opposite will equal zero.
This property is based on the basic rules of arithmetic and integer operations.
Let's break it down:

• Every integer has an opposite (or additive inverse).
• When a number and its opposite are added, they cancel each other out.
• The result is always zero.

For example, \(-18 + 18\ = 0\).
This property ensures that opposites neutralize each other.
It's useful for balancing equations and understanding the nature of numbers.