Problem 72 - Algebra A Combined Function

Question

Simplify each of the following. See Example 17. $$ -(-8.4) $$

Step-by-Step Solution

Answer

The expression \(-(-8.4)\) simplifies to \(8.4\).

Step 1: Understanding the Problem

The problem asks us to simplify the expression \(-(-8.4)\). We are dealing with two negative signs—it's important to understand how they interact with each other.

Step 2: Apply the Negative of a Negative Rule

The rule states that two negatives cancel each other out. When you have a negative outside of a bracket that encloses a negative number, the result is a positive. Therefore, \(-(-8.4)\) becomes \(+8.4\).

Step 3: Final Simplification

The simplified version of the expression \(-(-8.4)\) is \(8.4\), after applying the rule of negating a negative number.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero, often represented with a minus sign (-). They might seem tricky at first, but they play a crucial role in math.
Understanding how negative numbers work is essential, especially when simplifying expressions.
When you see a negative number, think of it as the opposite direction on the number line. For instance:

• Moving from zero to -3 is going three steps to the left.
• Conversely, moving from -3 to 0 requires three steps to the right.

These steps help to visualize negative numbers as simply mirroring positive numbers.
In our example, we dealt with two negatives:

• The first negative is in front of the bracket, while the second is inside of it.
• The two negatives together \( -(-8.4) \) result in a positive outcome, essentially flipping the negative inside to positive.

This flipping is a fundamental part of handling negatives effectively.

Simplifying Expressions

Simplifying expressions is a common task in algebra, and it’s more about "fewer, clearer steps" than manipulating numbers into a difficult operation.
The key is to make complex expressions easier to handle and understand.
When you simplify expressions:

• Always look for ways to combine similar parts or terms.
• Try to reduce the number of operations.
• Use arithmetic rules like the negative of a negative to streamline the expression.

In our original exercise, simplifying \( -(-8.4) \) involved using the rule of double negatives:

• We replaced an outer negative with a positive to simplify it to \( 8.4 \).

This process reduced a complex double-negative into a straightforward positive value.

Basic Algebra Rules

Basic algebra rules are the backbone of understanding mathematical expressions and equations. They are the "grammar" of math that ensures everyone understands the same thing when looking at an expression.
Some foundational rules include operations with variables and constants, managing expressions, and basic manipulations like distributing numbers across parentheses.One important rule used in our problem is the rule of "negative of a negative":

• This rule states that two negative signs cancel each other out to become a positive. \( -(-a) = a \)

Other key rules include:

• How to handle addition and subtraction with negatives, as they can change the outcome based on their placement.
• Distributing multiplication over addition or subtraction within parentheses: \( a(b + c) = ab + ac \)

Understanding these basic algebra rules allows us to simplify and solve equations more efficiently and accurately.