Problem 34 - Algebra 1

Question

Find the opposite of the number. $$-3$$

Step-by-Step Solution

Answer

The opposite of -3 is 3.

Step 1 - Understand what the opposite (or additive inverse) means

The opposite of a number is the value that the number has to be added to get the result as 0. In other words, it's the additive inverse of the number. It is the sign-switched version of that number.

Step 2 - Find the opposite of the given number

The given number is -3. To find the opposite, simply switch the sign. This means the opposite of -3 is 3.

Key Concepts

Additive Inverse

The concept of the additive inverse is a fundamental part of arithmetic and algebra. Understanding this can significantly help in solving equations and working with numbers. The additive inverse of a number is the number that, when added to the original number, equals zero. This means the two numbers "cancel" each other out due to their opposite signs.
For example, the additive inverse of -3 is 3. When you add -3 and 3 together, the result is zero:

• -3 + 3 = 0

To find the additive inverse of any number, simply change its sign. If the number is positive, its additive inverse is negative, and vice versa. It's a useful concept because it helps ensure balance in equations and expressions, playing a crucial role in solving them.

Integers

Integers are the set of whole numbers that include all positive numbers, negative numbers, and zero. They do not contain fractions or decimals.

• Examples of integers: ...,-3, -2, -1, 0, 1, 2, 3,...

Integers are everywhere in mathematics and everyday life. They help quantify and interpret data, allowing us to measure things accurately. Understanding integers is very important because they lay the foundation for many other math concepts.
When dealing with integers, you can conduct a variety of arithmetic operations such as addition, subtraction, multiplication, and division. Each of these operations adheres to specific rules, especially considering the signs of the numbers:

• Addition: A positive plus a positive is positive; a negative plus a negative is negative; if the signs differ, subtract and take the sign of the larger number.
• Subtraction: Transform into addition by adding the additive inverse of the subtracted integer.
• Multiplication and Division: Like signs yield a positive result; unlike signs yield a negative result.

Algebra Basics

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It allows us to express and solve equations and relationships using variables. A variable typically represents an unknown number and is often represented by letters like x, y, or z.
One of the basic principles of algebra is maintaining balance in equations, much like a scale. This is where the concept of the additive inverse becomes particularly useful. When solving for a variable, we often move terms around on both sides of an equation. To isolate a variable, you can add or subtract terms or even multiply and divide, as long as the same operation is performed on both sides of the equation.

• Example: If you have x - 3 = 0, adding 3 to both sides "undoes" the subtraction, leaving you with x = 3.

In algebra, learning to manipulate equations is key. The understandings of positive and negative numbers, represented in integer form, are foundational in organizing and working with expressions and equations efficiently.