Problem 19 - Fundamentals of Mathematics

Question

The sum of three consecutive integers is \(-27 .\) What are they?

Step-by-Step Solution

Answer

The integers are -10, -9, and -8.

Step 1: Define Variables

Let's define the three consecutive integers. If we call the first integer \( x \), then the next two consecutive integers would be \( x+1 \) and \( x+2 \).

Step 2: Set Up Equation

Given that the sum of these three consecutive integers is \(-27\), we can set up the equation: \[ x + (x + 1) + (x + 2) = -27. \]

Step 3: Simplify the Equation

Combine the like terms in the equation: \[ 3x + 3 = -27. \]

Step 4: Solve for x

First, subtract 3 from both sides of the equation: \[ 3x = -30. \]Then, divide both sides by 3: \[ x = -10. \]

Step 5: Determine the Consecutive Integers

Now that we know \( x = -10 \), the three consecutive integers are:- First integer: \( x = -10 \)- Second integer: \( x+1 = -9 \)- Third integer: \( x+2 = -8 \).

Key Concepts

Integer Operations

Integer operations are fundamental to solving problems involving consecutive integers. An integer is a whole number that can be positive, negative, or zero. In the context of adding integers, we perform operations much like simple arithmetic with straightforward rules.

• When you add two positive integers, you get a positive sum. For example, 3 + 4 equals 7.
• Adding two negative integers results in a negative sum. For example, -2 + -3 equals -5.
• Adding a positive integer and a negative integer is like subtracting one from the other—whichever number is larger in absolute value determines the sign of the result.

With consecutive integers like in this exercise, finding their sum involves simple addition of terms like \( x + (x+1) + (x+2) \). Note how integers follow each other in sequence.
Understanding these operations helps with setting up equations that accurately represent word problems involving sums of numbers.

Equation Setup

Setting up the equation correctly is an essential math skill, especially in word problems dealing with consecutive integers. The first step involves defining the variables accurately. Consider three consecutive integers: if the first one is represented by \( x \), then the others will logically be \( x+1 \) and \( x+2 \).
Next, translate the problem statement into a mathematical equation. In this exercise, it is stated that the sum of these integers is -27. This becomes your key equation: \( x + (x + 1) + (x + 2) = -27 \).
Notice how each integer differs by 1, reflecting their "consecutive" nature. This setup allows us to translate English into math, which is a critical skill for tackling complex word problems. Ensuring accuracy during this step will make solving the equation straightforward.

Solving Equations

Solving equations is about finding the unknown value that makes the equation true. This usually involves simplification and manipulation of the given expression. Let's simplify and solve the equation \( x + (x + 1) + (x + 2) = -27 \). We start by combining like terms:

• Add the \( x \) terms: \( x + x + x = 3x \).
• Combine numerical constants: \( 1 + 2 = 3 \).
• Thus, the equation becomes \( 3x + 3 = -27 \).

To isolate \( x \), follow the steps:
1. Subtract 3 from both sides to get \( 3x = -30 \). 2. Divide both sides by 3 to find \( x = -10 \).
Finally, substitute \( x = -10 \) back into the expressions for consecutive integers: \(-10, -9, -8\). The solution is thus complete. Solving requires patience and carefully following each step for accuracy.