Problem 84 - College Algebra

Question

State the name of the property illustrated. \((x+4)+[-(x+4)]-0\)

Step-by-Step Solution

Answer

The properties illustrated in the expression \((x+4)+[-(x+4)]-0\) are the 'Additive Inverse Property' and the 'Identity Property of Addition'.

Step 1: Identify the Operations and Elements Involved

We first need to identify what operations and elements the expression consists of. Looking at the expression \((x+4)+[-(x+4)]-0\), we see it involves addition and subtraction of an element \(x+4\) and it's corresponding additive inverse \(-(x+4)\). The operation is also along with subtraction of \(0\).

Step 2: Recognize the Additive Inverse Property

The additive inverse property states that the sum of a number and its additive inverse (its negation) is always zero. This implies that any number plus its additive inverse equals zero, represented as \(a + (-a) = 0\). In the expression, \(x+4\) and \(-(x+4)\) are additive inverses of each other, thus their sum is \(0\).

Step 3: Recognize the Identity Property of Addition

The Identity Property of Addition states that the sum of any number and zero is the number itself, represented as \(a + 0= a\). In the expression after the second step, since the sum of \((x+4)\) and \((-(x+4))\) is zero, the expression simplifies to \(0 - 0\), and according to the identity property of addition, this is \(0\).

Key Concepts

Identity Property of Addition

The Identity Property of Addition is a fundamental concept in mathematics that ensures stability when adding numbers. Simply put, it means that adding zero to any number does not change the value of that number. This property is represented by the equation \( a + 0 = a \). The number zero is known as the 'additive identity' because it does not alter the value it is added to.

Think of it as a way to verify that the original number remains intact, even with the presence of zero. This property simplifies calculations and proves extremely useful when solving equations. In our expression \((x+4) + [-(x+4)] - 0\), after we apply the additive inverse property, we are left with \(0 - 0\). Here, using the identity property, we see that subtracting zero from zero will still yield zero, confirming the stability of zero as an additive identity.

Addition and Subtraction

Addition and subtraction are inverse operations that help us move numbers to balance equations or solve for variables. Subtraction is often seen as adding a negative, which aligns perfectly with concepts like the additive inverse. When we think about the expression \((x+4) + [-(x+4)] - 0\), it's important to dissect what each part is doing.

In this case, \((x+4)\) is added to its negative \(-(x+4)\), which is equivalent to \(x+4\) being subtracted by \(x+4\). Thanks to the additive inverse property, this part of the expression equals zero.

Subtracting zero afterwards does not change the value, further supported by the concept of subtraction as the opposite operation of addition. Such operations are crucial for simplifying expressions and solving algebraic equations effectively.

Expressions Simplification

Simplifying expressions involves reducing them to their simplest form without changing the mathematical value. The goal is to make calculations easier and results more understandable. In the expression \((x+4) + [-(x+4)] - 0\), several properties help achieve this simplification.

Firstly, the additive inverse property is applied when \(x+4\) is paired with \(-(x+4)\), bringing us immediately to zero. This step demonstrates how opposing terms can cancel each other.

Secondly, the subtraction of zero is addressed by the identity property of addition, indicating no further change to the simplified form of zero.

• Recognizing which terms can be cancelled or simplified is a skilful part of algebra.
• Use properties like inverse and identity to guide through simplification.

In conclusion, understanding these properties and operations makes it easier to handle and simplify mathematical expressions effortlessly.