Problem 9 - Elementary Algebra

Question

Find the opposite of each real number. $$ -6 $$

Step-by-Step Solution

Answer

Answer: 6

Step 1: Identify the given number

The given number is -6.

Step 2: Change the sign of the given number

Since the given number is negative (-6), the opposite will be positive. To find the opposite, change the negative sign (-) to a positive sign (+): $$-(-6)$$

Step 3: Write the final answer

The opposite of -6 is: $$+6$$

Key Concepts

Real Numbers

Real numbers are the set of all numbers that can be found on the number line. This includes all the positive and negative integers, fractions, and decimals. They can be as small as negative infinity and as large as positive infinity, encompassing all possible values that can be used for counting, measuring, and continuous calculations.

To picture them, imagine a straight line where every point corresponds to a real number. This number line is fundamental in understanding many concepts in algebra, including the idea of opposite numbers. When we work with real numbers, we use them in various algebraic operations which allows us to solve equations and model real world situations.

Opposite Numbers

The concept of opposite numbers is quite simple – it’s about finding a number that, when added to the original number, yields zero. The opposite of a number is located at the same distance from zero on the number line, but in the opposite direction. For any real number, you can find its opposite by changing its sign.

For example, the opposite of -6 is +6. Even in practical terms, if you have a debt of \(6, the opposite scenario would be having a credit of \)6. Opposite numbers play a crucial role in understanding balance and symmetry in mathematics, and they are key in solving algebraic equations.

Algebraic Operations

Algebraic operations are the bread and butter of algebra. They include addition, subtraction, multiplication, and division, but they also extend to more complex operations such as exponentiation, and finding roots. When working with real numbers, these operations follow specific properties like commutativity, associativity, and distributivity.

Understanding how to use these operations is critical when solving for unknowns in equations, simplifying expressions, and in performing operations with opposite numbers. For instance, subtracting a number is the same as adding its opposite. This concept is not only fundamental in algebra but also in a broad range of mathematical topics and real-world problem solving.