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Chapter 2: Problem 47

Solve each problem imolving consecutive integers. Find three consecutive odd integers such the sum of the least integer and the middle integer is 19 more than the greatest integer.

Short Answer

Expert verified
The three integers are 21, 23, and 25.

Step by step solution

01

Define the Variables

Let the three consecutive odd integers be represented by the variables Let the first (least) odd integer be: \( x \)Let the second (middle) odd integer be: \( x + 2 \)Let the third (greatest) odd integer be: \( x + 4 \)
02

Set Up the Equation

According to the problem, the sum of the first and middle integers is 19 more than the greatest integer. Therefore, set up the equation: \( x + (x + 2) = (x + 4) + 19 \)
03

Simplify the Equation

Combine like terms on the left side and then simplify: \( 2x + 2 = x + 23 \)
04

Solve for x

Isolate the variable x by subtracting x from both sides: \( 2x + 2 - x = 23 \) This simplifies to: \( x + 2 = 23 \) Then subtract 2 from both sides: \( x = 21 \)
05

Find the Three Integers

Now that the value of x is found, the three odd integers can be calculated as follows:First integer: \( x = 21 \)Second integer: \( x + 2 = 23 \)Third integer: \( x + 4 = 25 \)
06

Verify the Solution

Ensure that the sum of the least integer and the middle integer is 19 more than the greatest integer:\( 21 + 23 = 44 \)\( 25 + 19 = 44 \)Since both sides are equal, the solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

algebraic equations
Algebraic equations are mathematical statements that show the equality between two expressions. They often contain variables, which represent unknown values. To solve an algebraic equation, you need to find the value of these variables that makes the equation true. In our exercise, we defined the consecutive odd integers with variables and set up an equation based on the given problem. This approach helps to systematically solve for the unknowns by applying algebraic principles such as combining like terms and isolating the variable.
consecutive numbers
Consecutive numbers follow one another in an unbroken sequence. In math, when we talk about consecutive integers, they follow directly from one another without gaps. For example, 3, 4, 5 are consecutive numbers. In our problem, we deal with consecutive odd integers, which means each number is two units apart from the next, such as 21, 23, and 25. Understanding this sequence is crucial because it helps in setting up correct algebraic expressions for solving the problem.
problem-solving steps
Effective problem-solving steps in algebra involve several key stages. First, you need to understand the problem and identify what is being asked. Next, define the variables that represent the unknowns. Write down an equation based on the problem statement. Simplify and solve the equation systematically. Finally, verify your solution by checking if it satisfies the original problem conditions. In our exercise, we defined the integers, set up the equation based on the relationship given in the problem, solved for the variable, and then verified our solution.
integer properties
Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. Some important properties of integers include their distribution on the number line and their ability to be added, subtracted, multiplied, and divided. Consecutive integers, like the ones in this problem, have a special relationship where each number differs by a fixed amount (usually one). Understanding integer properties helps in solving algebra problems involving sequences and recognizing consistent patterns. In the current exercise, knowing that consecutive odd integers differ by 2 simplifies setting up the equation and solving it step by step.

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