Problem 4 - Fundamentals of Mathematics

Question

The sum of three consecutive even integers is six more than four times the middle integer. Find the integers.

Step-by-Step Solution

Answer

The integers are -8, -6, and -4.

Step 1: Define Variables

Let's call the three consecutive even integers as \( x \), \( x+2 \), and \( x+4 \). Here, \( x \) is the first even integer, \( x+2 \) is the middle integer, and \( x+4 \) is the third even integer.

Step 2: Set Up Equation

We know that the sum of these three integers is six more than four times the middle integer. So, we can write the equation: \[ x + (x + 2) + (x + 4) = 4(x + 2) + 6 \] Simplifying the left side, we get \[ 3x + 6 = 4(x + 2) + 6 \]

Step 3: Simplify Equation

Now, let's simplify the equation:Expand the right side: \[ 4(x + 2) + 6 = 4x + 8 + 6 \] Which simplifies to: \[ 4x + 14 \] So the equation becomes: \[ 3x + 6 = 4x + 14 \]

Step 4: Solve for x

To solve for \( x \), we'll rearrange the equation:Subtract \( 3x \) from both sides:\[ 6 = x + 14 \]Subtract 14 from both sides:\[ x = -8 \].

Step 5: Find the Integers

Now, substitute \( x = -8 \) back into the expressions for the integers:The first integer: \( x = -8 \)The second integer: \( x + 2 = -6 \)The third integer: \( x + 4 = -4 \).

Key Concepts

Consecutive Integers

When dealing with consecutive integers, it's essential to understand what the term "consecutive" means. Consecutive integers are numbers that follow each other in a sequence, where each number is one more than the previous number. When speaking about consecutive even integers, the pattern is slightly different.

• Even consecutive integers always have a difference of 2 between them because even numbers by definition are divisible by 2.
• This means if you start with an even number, the next even in the sequence would be that number plus 2.
• For example, if our first consecutive even integer is 4, the next two would be 6 and 8.

To express them algebraically, we use variables. In our exercise, we used:

• First integer: \( x \)
• Second integer: \( x + 2 \)
• Third integer: \( x + 4 \)

This method helps when setting up and solving equations to find unknown integers.

Solving Equations

Solving equations is the process of finding the unknown values that satisfy a given mathematical statement. In our exercise, the equation represented the condition: "The sum of three consecutive even integers is six more than four times the middle integer."
To solve this, we created the equation:\[x + (x + 2) + (x + 4) = 4(x + 2) + 6\]Simplifying the equation gives us:

• Combine like terms on the left: \( 3x + 6 \)
• Expand and simplify the right: \( 4x + 14 \)

The next step is to solve for \( x \):

• Subtract \( 3x \) from both sides to isolate terms involving \( x \)
• Subtract 14 from both sides to find that \( x = -8 \)

One essential step in solving equations is maintaining balance, which means whatever operation you do to one side of the equation, you must do to the other side as well.This ensures the equality remains true.

Mathematical Expressions

Mathematical expressions are a combination of numbers, variables, and operators (like addition or multiplication) that represents a particular value or concept. In our task, understanding expressions is key to setting up and resolving the equation at hand.
In the exercise, the mathematical expressions included:

• \( x \), \( x+2 \), \( x+4 \): Represent the three consecutive integers
• \( 4(x+2) + 6 \): Represents four times the middle integer plus six

Expressions help us communicate mathematical ideas and relationships without complete sentences. When dealing with problems involving these, it's crucial to:

• Translate words into mathematical symbols correctly.
• Understand operator precedence, that is, which operations to perform first, often involving parentheses.

By setting up and manipulating these expressions, you can make sense of complex problems and find the solutions. Recognizing the relationships between different expressions is a powerful tool in algebra.