Problem 78 - Algebra A Combined Function

Question

Write each algebraic expression described. Simplify if possible. See Example \(11 .\) If \(x\) is the first of three consecutive even integers, write their sum as an algebraic expression in \(x\).

Step-by-Step Solution

Answer

The sum of three consecutive even integers expressed in terms of \( x \) is \( 3x + 6 \).

Step 1: Understand Consecutive Even Integers

Consecutive even integers have a difference of 2 between each number. If the first integer is denoted as \( x \), then the next two consecutive even integers would be \( x+2 \) and \( x+4 \).

Step 2: Write the Expression for the Sum

To find the sum of these three integers, write down the expression: \[ x + (x + 2) + (x + 4) \] This represents the sum of the three consecutive even integers.

Step 3: Simplify the Algebraic Expression

Simplify the expression by combining like terms: \[ x + (x + 2) + (x + 4) = x + x + 2 + x + 4 = 3x + 6 \]The simplified expression for the sum of three consecutive even integers is \( 3x + 6 \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are foundational in algebra, allowing us to represent relationships and solve problems in a generalized form. In this exercise, we deal with an algebraic expression representing the sum of consecutive even integers. Here, the expression uses the variable \(x\) to denote numbers. This enables us to handle arithmetic involving unknown quantities.

• The term "consecutive" refers to numbers that follow each other in order without gaps, specifically even numbers in this context.
• By manipulating the variable \(x\), we can create expressions to describe sequences or series of numbers and their properties.

Understanding algebraic expressions helps in structuring mathematical equations that are flexible and can be applied to solve problems involving patterns or predictable sequences of numbers.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a critical skill in mathematics, as it makes complex problems more manageable and easier to solve.

• In our example, the initial expression for the sum of three consecutive even integers is \( x + (x + 2) + (x + 4) \).
• To simplify this expression, we combine like terms, which are terms that contain the same variable raised to the same power. Here, the like terms are the \(x\)'s.

Simplification entails merging these like terms: \(x + x + x = 3x\). Now, we add the constant numbers: \(2 + 4 = 6\).

This results in the simplified expression \(3x + 6\), which maintains the original mathematical properties but is easier to work with. Through simplification, complex expressions become more concise, making it simpler to identify solutions and understand the relationships between numbers.

Exploring Even Integers

Even integers are numbers that can be exactly divided by 2. They follow a regular pattern, significant in various mathematical sequences and calculations.

• An even integer is usually of the form \(2n\), where \(n\) is an integer.
• In practical problems involving consecutive even numbers, if the first number is \(x\), then the next two consecutive even numbers are \(x + 2\) and \(x + 4\).

This pattern means there is always a consistent difference of 2 between any two consecutive even integers. Recognizing this pattern is crucial when solving problems that involve series or sequences of even numbers.

Having this understanding helps create and simplify algebraic expressions for various applications in mathematics, particularly when examining sums or other operations involving consecutive numbers.