Problem 12 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

Associative property of addition.

Step 1: Understand the Problem

We analyze the given problem to determine what is being asked.

Step 2: Set Up the Solution

The statement \((x+3)+(-3)=x+[3+(-3)]\) is showcasing a regrouping or rearrangement of terms during addition without changing their order. This is the hallmark of the associative property of addition. It specifies that no matter how the numbers are grouped in an addition operation, the result is the same. Here, the parentheses are moved to group \(x\) with \(3+(-3)\) instead of \((x+3)\) with \(-3\), indicating the use of the associative property.

Step 3: Solve the Problem

Applying the relevant mathematical techniques, we work through the solution step by step.

Step 4: State the Result

Associative property of addition.

Key Concepts

Addition

Addition is one of the fundamental mathematical operations where two or more numbers (or expressions) are combined to make a larger number. It's vital in everyday life and is widely used in more advanced mathematics. Understanding addition simplifies dealing with numbers, whether you're calculating a grocery bill or balancing a checkbook. In this exercise, addition is used in an algebraic expression, meaning instead of just numbers, you have unknowns, like \(x\). The crucial takeaway in addition is that the order of the numbers doesn't matter, thanks to the associative and commutative properties. You can add numbers in any sequence or group them in different ways, and the sum will remain the same. This flexibility makes computation easier and more efficient.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like \(x\)), and operation symbols. They are the building blocks of algebra and allow you to describe mathematical situations and relationships. When you encounter an algebraic expression, you're looking at a combination of terms. For example, \((x + 3) + (-3)\) is an algebraic expression with three terms. Understanding how to manipulate expressions, like using the associative property, helps simplify them for easier problem-solving. When dealing with expressions, it's essential to properly apply properties of operations to rearrange or simplify without altering their value. This keeps the integrity of the expression intact while solving it or transforming it into a different form. Learning to skillfully handle algebraic expressions is a gateway to more advanced topics, like equations and functions, which are central to algebra and calculus.

Properties of Operations

The properties of operations, such as associative, commutative, and distributive, are fundamental rules used in math to simplify and solve expressions and equations.

• Commutative Property: This suggests that the order in which you add or multiply numbers doesn't affect the outcome. For addition, \(a + b = b + a\).
• Associative Property: This assures us that how numbers are grouped in an operation doesn't change their total. For addition, \((a + b) + c = a + (b + c)\). This allows flexibility in computation or rewriting expressions.
• Distributive Property: It combines addition and multiplication. This property lets us distribute a multiplied term across terms in parentheses, such as \(a(b + c) = ab + ac\).

These properties are powerful tools in algebra and are essential for rewriting expressions, solving equations, and simplifying complex mathematical statements. By mastering them, a student enhances their ability to tackle various mathematical challenges with confidence and efficiency.