Problem 58 - College Algebra

Question

state the name of the property illustrated. $$ (x+4)+[-(x+4)]=0 $$

Step-by-Step Solution

Answer

The property illustrated is the Additive Inverse Property.

Step 1: Identify the expression

We first identify the expression \( (x+4) \) and it's mirror image which is present in the negative form, \(- (x+4) \). The minus sign indicates the opposite of whatever is inside the brackets.

Step 2: Simplify the equation

Adding \((x+4)\) and \(- (x+4)\), we get 0. This is because \(x+4\) and \(-x-4\) cancel each other out.

Step 3: Identify the Property

The principle that is being demonstrated here is known as the Additive Inverse Property. This property states that for any number \(x\), if we add it's negative equivalent \(-x\), the result is always 0.

Key Concepts

Understanding Algebraic Properties

Algebraic properties are fundamental rules that govern how mathematical operations are carried out. These properties help ensure consistency and accuracy when performing algebraic calculations. In this exercise, we're focusing on one specific algebraic property known as the Additive Inverse Property. This is just one of several important properties used in algebra, including:

• Commutative Property: Order doesn't matter in addition or multiplication, expressed as \(a + b = b + a\).
• Associative Property: Grouping doesn't affect the result in addition or multiplication, such as \((a + b) + c = a + (b + c)\).
• Distributive Property: Multiplying a sum by a number equals multiplying each addend by the number and then adding the products, i.e., \(a(b + c) = ab + ac\).

In our exercise, the Additive Inverse Property is the star of the show, demonstrating how each number has an opposite that results in zero when both are added together.

Exploring the Addition of Integers

Addition is a basic operation in mathematics, but when it comes to integers, there's more to it than just adding numbers. Integers include not only positive numbers but also their negative counterparts and zero. When dealing with the addition of integers, certain rules and properties come into play to maintain a balance:

• Adding Positive Integers: The result is typically a larger positive integer. For instance, \( 3 + 2 = 5 \).
• Adding Negative Integers: The result remains negative. Example: \(-3 + (-2) = -5\).
• Adding a Positive and a Negative Integer: It's like finding their difference and considering the sign of the bigger number. For example, \(5 + (-3) = 2\).

In the given exercise, we see the addition of the expression \((x + 4)\) and its negative counterpart \(- (x + 4)\), which showcases how the distinct properties of integer addition lead to a sum of zero, illustrating the Additive Inverse Property in action.

Inverse Operations and Their Role

Inverse operations are pairs of operations that undo each other. In the context of addition, the inverse operation is subtraction. However, when looking at the Additive Inverse, it's a bit more specific:

• Additive Inverse: For any number \(x\), its additive inverse is \(-x\). Together, when they are added, they result in zero: \(x + (-x) = 0\).
• Utility in Algebra: The concept of inverse operations helps to solve equations, simplify expressions, and maintain equation balance.

Inverse operations are not just limited to addition and subtraction. They apply to multiplication and division as well. For instance, the multiplicative inverse of a number \(x\) is \(1/x\), since \(x \, \cdot \, 1/x = 1\). By understanding inverse operations, you develop the skills to tackle more complex algebraic challenges systematically. The exercise at hand excellently shows how understanding and applying inverse operations can simplify a problem and reveal solutions elegantly.