Problem 35 - Elementary Algebra

Question

Rewrite the problem in a simpler form. $$ -(-4) $$

Step-by-Step Solution

Answer

Question: Simplify the expression -(-4). Answer: 4

Step 1: Understand the double negative

In mathematics, a double negative ("-") cancels out with another negative. When you see two negative signs together, like in this expression -(-4), you can simplify this expression by removing both negative signs.

Step 2: Remove the double negative

Since the two negative signs cancel each other out, we can rewrite the expression without the negative signs. So, -(-4) becomes simply: $$ 4 $$

Step 3: Write the final simplified expression

Now that the double negative has been removed, the expression is simplified as: $$ 4 $$

Key Concepts

Simplifying Algebraic Expressions

When it comes to simplifying algebraic expressions, the goal is to make them as straightforward as possible. This involves combining like terms, using the distributive property when necessary, and eliminating any double negatives. In the context of the given exercise, the expression is very straightforward, consisting of a double negative which simplifies to a positive number.

For instance, when you encounter an expression such as \( -(-x) \), you can interpret the double negative as the opposite of negative, which translates to positive. Hence, the expression simplifies to \( x \). This principle holds true no matter the complexity of the expression. By applying this rule consistently, along with other algebraic principles, you can simplify even the most daunting algebraic expressions into a more manageable and understandable form.

Negative Numbers

Negative numbers are an essential part of algebra and the number system as a whole. It's important to understand how they interact with various operations. For example, the multiplication or division of two negative numbers results in a positive number. On the other hand, if you multiply or divide a positive number with a negative one, the result is negative.

In the exercise \( -(-4) \), we exclusively deal with negative numbers. The expression represents a number, -4, being multiplied by -1, since any number preceded by a minus sign can be considered as being multiplied by -1. This results in a positive value because, as per the rules governing negative numbers, a negative multiplied by a negative gives a positive. Understanding the behavior of negative numbers allows for accurate simplification of expressions and equations involving negative values.

Basic Algebraic Operations

The foundation of algebra lies in the understanding of basic operations such as addition, subtraction, multiplication, and division. These operations often work together with negative numbers and the concept of simplifying algebraic expressions. In the given exercise, the operation is quite simple: we are removing the double negative, which is akin to multiplying two negative numbers. As mentioned earlier, this results in a positive number.

The ability to carry out basic algebraic operations fluently is crucial, as it can determine the correctness of the simplification process. A solid grasp of these operations also enables students to move on to more complex algebraic concepts with confidence. For any beginner, it is highly recommended to practice these basic operations with both positive and negative numbers to develop a strong algebraic foundation.