Problem 48 - Algebra for College Students

Question

Solve each problem involving consecutive integers. Find three consecutive even integers such that the sum of the least integer and the greatest integer is 12 more than the middle integer.

Step-by-Step Solution

Answer

The three consecutive even integers are 10, 12, and 14.

Step 1 - Define the integers

Let the three consecutive even integers be represented by: Let the first even integer be x. The second consecutive even integer will then be x+2. The third consecutive even integer will be x+4.

Step 2 - Set up the equation

The problem states that the sum of the least integer and the greatest integer is 12 more than the middle integer. Set up an equation based on this information: x + (x+4) = (x+2) + 12.

Step 3 - Simplify and solve the equation

Combine like terms and solve for x: x + (x+4) = (x+2) + 12 2x + 4 = x + 14 Subtract x from both sides: x + 4 = 14 Subtract 4 from both sides: x = 10.

Step 4 - Find all integers

Using the value of x, find the three consecutive even integers: The first even integer is 10. The second even integer is 10 + 2 = 12. The third even integer is 10 + 4 = 14.

Step 5 - Verify the solution

Verify if the integer values satisfy the condition given in the problem: The sum of the first and the third integers is 10 + 14 = 24. The middle integer is 12, and 12 more than the middle integer is 12 + 12 = 24. Since both sums are equal, the integers (10, 12, 14) are the correct solution.

Key Concepts

even integers

Even integers are numbers that are divisible by 2. They include numbers like 2, 4, 6, 8, and so on. To find consecutive even integers, you add 2 to the previous even integer. For example, after 2 comes 4, then 6, and so forth. In our problem, we defined the integers as follows:

• The first even integer is denoted as x.
• The next even integer is x + 2.
• The next even integer is x + 4.

Understanding even integers is crucial because it helps you set up the problem accurately.

algebraic equations

Algebraic equations involve numbers and variables combined using mathematical operations. In this problem, we set up an equation to relate the even integers. The equation given was: \[ x + (x+4) = (x+2) + 12\]
Here, each term represents either an integer or a combination of integers. Setting up and simplifying algebraic equations helps you find unknown values, which is the goal of this problem.

solving equations

Solving equations involves finding the value of unknowns that make the equation true. In our problem, we solved for x:

• Combine like terms: \( x + (x+4) = (x+2) + 12 \) turns into \( 2x + 4 = x + 14 \)
• Isolate the variable: Subtract x from both sides to get \( x + 4 = 14 \).
• Simplify: Subtract 4 from both sides to find \( x = 10 \).

By solving the equation, we found that the first even integer is 10.

integer properties

Integers are whole numbers that can be positive, negative, or zero. Even integers are a subset of integers. They have properties that make solving problems straightforward:

• Consecutive integers follow a set pattern.
• Adding or subtracting integers retains another integer (e.g., 2 + 2 = 4).

Our understanding of integers allowed us to verify our solution correctly, ensuring that 10, 12, and 14 meet the given conditions.

problem-solving strategies

Effective problem-solving strategies help you find solutions systematically. In this problem, we used the following strategies:

• Define variables: We represented the integers as x, x+2, and x+4.
• Set up equations: Based on the problem statement, we formulated an equation.
• Simplify: We combined like terms and solved for x.
• Verify: After finding the integers, we checked our solution to ensure it met all conditions.

Following these steps, we efficiently solved for the consecutive even integers and verified our answer.