Problem 34 - Fundamentals of Mathematics

Question

Find four consecutive integers that add to negative two.

Step-by-Step Solution

Answer

The integers are -2, -1, 0, and 1.

Step 1: Define the Integers

Let the first integer be \( x \). Then the four consecutive integers can be expressed as \( x, x+1, x+2, \) and \( x+3 \).

Step 2: Write the Equation

Since the sum of these integers is given to be -2, we can write the equation: \( x + (x+1) + (x+2) + (x+3) = -2 \).

Step 3: Simplify the Equation

Simplify the left side of the equation: \( 4x + 6 = -2 \).

Step 4: Solve for \( x \)

Subtract 6 from both sides to get \( 4x = -8 \). Divide both sides by 4 to find \( x = -2 \).

Step 5: Identify the Integers

Now that we know \( x = -2 \), the four consecutive integers are \( -2, -1, 0, \) and \( 1 \).

Step 6: Verification

Check that the sum of these integers is -2: \( -2 + (-1) + 0 + 1 = -2 \). The condition is satisfied.

Key Concepts

Integer Equations

Integer equations are equations that involve integers, which are whole numbers that can be positive, negative, or zero. In the context of this exercise, we are dealing with consecutive integers, which are integers that follow each other in sequence. For four consecutive integers, these can be represented as \( x, x+1, x+2, \) and \( x+3 \). This representation helps in setting up an equation to solve problems involving these sequences. By translating a word problem into an integer equation, we can find the unknown variable, typically represented by \( x \).

Here are some key pointers about integer equations:

• Integer equations consist of terms connected by addition, subtraction, multiplication, or division.
• Variables like \( x \) represent unknown integers.
• Consecutive integers can be systematically represented using the starting integer followed by increments (e.g., \( x, x+1, x+2 \)).

This structure makes it easy to solve problems by transforming words into a mathematical form.

Solving Linear Equations

Solving linear equations is a fundamental algebraic process that involves finding the value(s) of the unknown variable(s) that make the equation true. In our problem, solving involves simple operations to isolate \( x \). The equation from our exercise, \( 4x + 6 = -2 \), is a classic example of a linear equation.

Here's a simple rundown on solving such linear equations:

• First, simplify both sides of the equation if necessary. Combine like terms.
• Next, use inverse operations to isolate the variable. This means you perform operations that will break down the equation systematically.
• In our example, we subtracted 6 from both sides, giving \( 4x = -8 \).
• Then, divide by the coefficient next to the variable (in this case, 4) to solve for \( x \).
• This approach yields \( x = -2 \) as the solution.

Each step brings the equation closer to the solution by simplifying or reorganizing it, making it easier to determine the value of \( x \).

Problem Solving Steps

Effective problem solving involves a clear step-by-step approach that ensures you don't miss any critical part of the equation. Here’s a breakdown of the problem solving steps used in this exercise:

1. **Define the Variables:** Identify what you are solving for and represent unknowns with variables. In this problem, we defined the first integer as \( x \).
2. **Translate the Problem:** Convert the word problem into a mathematical equation based on the definition of the variables. We wrote \( x + (x+1) + (x+2) + (x+3) = -2 \).
3. **Simplify and Solve:** Simplify the equation by combining like terms and using inverse operations to solve for the variable. We combined terms to get \( 4x + 6 = -2 \) and solved for \( x \).

• This involves getting \( 4x = -8 \) by subtracting 6 and dividing to find \( x = -2 \).

4. **Validate the Solution:** Always revisit the original problem to validate your solution. For instance, we calculated \( -2, -1, 0, \) and \( 1 \) to ensure their sum is indeed \(-2\).
These steps not only help in math but develop a structured approach to tackling problems in any field, ensuring no detail is overlooked.