Problem 19 - Algebra and Trigonometry

Question

Rewrite the expression using the given property of real numbers. Commutative Property of Addition, \(\quad x+3=\) ________

Step-by-Step Solution

Answer

\(x + 3\) can be rewritten as \(3 + x\).

Step 1: Identify the Property

The exercise asks us to apply the Commutative Property of Addition to the expression. This property states that the order of numbers in addition does not affect the sum. In other words, for any numbers \(a\) and \(b\), \(a + b = b + a\).

Step 2: Apply the Commutative Property

Given the expression \(x + 3\), we can rearrange the terms using the Commutative Property. According to this property, \(x + 3\) is equal to \(3 + x\).

Step 3: Provide the Rewritten Expression

By applying the Commutative Property of Addition, the expression \(x + 3\) can be rewritten as \(3 + x\).

Key Concepts

Real Numbers

When dealing with algebra, we often encounter the term "real numbers." Real numbers include all the numbers we usually think of when we count, measure, or solve equations. This includes:

• Whole numbers (like 0, 1, 2,...)
• Integers (which are negative numbers, zero, and positive numbers)
• Rational numbers (like fractions and decimals that end or repeat)
• Irrational numbers (like √2 or π, which can't be written as a simple fraction)

Real numbers form a continuous line on the number line with no gaps. They are the building blocks of algebra and many other areas in mathematics. Understanding real numbers helps us comprehend operations such as addition, subtraction, multiplication, and division across a wide range of possible values.

Addition Property

The Commutative Property of Addition is a central concept when working with numbers. It tells us that the order in which two numbers are added does not change their sum. For example, if you have 5 + 8 or 8 + 5, both expressions equal 13. This property helps simplify computations and is foundational to understanding algebraic manipulations.When applying the Commutative Property, you can:

• Rearrange numbers to make calculations easier
• Verify that sums are consistent, regardless of order
• Swap variables in algebraic expressions without affecting results

In the exercise, the expression \(x + 3\) was rewritten as \(3 + x\) using this property. Recognizing this operation is key to simplifying complex expressions and solving equations more fluidly.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. In the expression \(x + 3\), \(x\) is a variable, and 3 is a constant number. Understanding algebraic expressions is crucial for solving equations and understanding relationships between numbers in algebra.Here's what to know about algebraic expressions:

• They represent general forms of addition, subtraction, and more
• Variables can stand for unknown or changeable values
• By manipulating expressions, we can solve problems involving unknowns

Using properties like the Commutative Property simplifies these expressions, making them more manageable. Understanding how to rearrange terms while maintaining equivalency is a fundamental skill in algebra.