Problem 1 - Algebra for College Students

Question

State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$[6+(-2)]+4=6+[(-2)+4]$$

Step-by-Step Solution

Answer

The associative property of addition justifies the statement.

Step 1: Identify the Given Expression

The expression provided is \([6 + (-2)] + 4 = 6 + [(-2) + 4]\). We need to determine the property that justifies this mathematical statement.

Step 2: Recognize the Property

Observe that the expression involves changing the grouping of numbers being added without changing the order of the numbers. This is an indicator of the Associative Property of Addition.

Step 3: Define the Associative Property

The Associative Property of Addition states that the way in which numbers are grouped in addition does not affect their sum. It indicates that \((a + b) + c = a + (b + c)\) for any numbers \(a\), \(b\), and \(c\).

Step 4: Apply the Associative Property

In the expression \([6 + (-2)] + 4 = 6 + [(-2) + 4]\), we see that the numbers are 6, -2, and 4. The grouping within the brackets shifts from \([6 + (-2)]\) to \([(-2) + 4]\), demonstrating the associative property, as the addition remains the same.

Key Concepts

Commutative Property of Addition

The commutative property of addition is a fundamental rule that helps in rearranging numbers to simplify calculations. This property tells us that changing the order of numbers being added does not alter the result. For example, if you have two numbers, like 3 and 5, adding them in any order will yield the same result:

• \(3 + 5 = 5 + 3\)
• Both expressions equal 8.

This property is highly practical when solving arithmetic problems, as it allows flexibility in computations.
You can swap the numbers to make calculations more convenient, especially in mental math.
Remember that it strictly applies to addition (and multiplication), not subtraction or division.

Properties of Addition

Addition is governed by a set of rules that help in performing calculations more efficiently. Three of the most significant ones include:

• Associative Property: This involves grouping numbers differently without changing their sum. Example: \((a + b) + c = a + (b + c)\).
• Commutative Property: As discussed, it reflects changing the order of numbers, \(a + b = b + a\).
• Identity Property: This indicates adding zero to any number keeps that number unchanged, \(a + 0 = a\).

Understanding these properties allows for strategic rearrangement of expressions to simplify and solve math problems.
They play a crucial role in algebra and everyday arithmetic.

Mathematical Justification

Mathematical justification involves explaining why certain mathematical rules and properties apply to an expression or equation. It's about providing a logical reasoning or proof for a solution:- Consider the original exercise statement \([6+(-2)]+4=6+[(-2)+4]\). Here, the justification involves the associative property since the grouping of numbers is changed.- Justifying the use of a property means showing that the solution follows specific mathematical rules and principles.
This ensures that conclusions drawn from an expression or equation are correct and verifiable.
Providing a mathematical justification is essential in confirming the accuracy of solutions and in understanding the 'why' behind a mathematical process.