Problem 4 - 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. $$ 1(x)=x $$

Step-by-Step Solution

Answer

Multiplicative identity property.

Step 1: Identify the Operation

The operation involved here is multiplication. The expression is presented as multiplication of the number 1 by any variable \(x\).

Step 2: Recognize the Property

In multiplication, the number 1 is known as the multiplicative identity. The property states that multiplying any number by 1 doesn't change the value of that number.

Step 3: State the Property

Therefore, the property that justifies the statement \(1(x) = x\) is the multiplicative identity property.

Key Concepts

Commutative Property of Addition

The commutative property of addition is a fundamental idea in mathematics. It tells us that when adding two numbers, we can switch the order of the numbers and still get the same result. This property is essential for simplifying math problems and confirming that the order of addition doesn't matter. For instance, if we have the numbers 3 and -4, we can write:

• 3 + (-4) = -4 + 3

This illustrates that regardless of the order, the sum remains the same.
Why is this important? Understanding this property helps improve our mental math skills. We can rearrange numbers to make calculations easier.
So, next time you need to simplify an addition problem, remember you can switch the numbers around! It showcases the flexibility of numbers when they are added.

Algebraic Properties

Algebraic properties are rules that apply to numbers and variables alike which help solve equations. These rules include properties like the commutative, associative, distributive, and identity properties. Each property serves a unique purpose:

• **Commutative Property** - changing the order doesn't change the sum or product.
• **Associative Property** - changing the grouping doesn't change the sum or product.
• **Distributive Property** - allows us to multiply a single term across terms inside a parenthesis.
• **Identity Property** - involves the identity elements such as 0 for addition and 1 for multiplication.

These properties simplify complex equations and enable us to solve equations systematically.
Fully grasping these concepts empowers us to handle more difficult algebraic expressions and ensures consistent results in calculations.

Multiplication in Algebra

Multiplication in algebra involves multiplying numbers and variables. A key concept here is the multiplicative identity property. This property states that any number multiplied by 1 remains unchanged. Recognizing this helps simplify equations and solutions.
Let's consider an example like the statement:

• 1(x) = x

This means multiplying x by 1 leaves x the same; x 'keeps its identity' — hence the term 'identity property'.
This concept is crucial when rearranging equations or solving for variables, making sure you do not alter the values unintentionally. Understanding multiplication in algebra is necessary for handling equations effectively and simplifying algebraic expressions quickly.