Problem 32 - Elementary Algebra

Question

Find the sums. \(14+(-6)\)

Step-by-Step Solution

Answer

Answer: The sum of 14 and -6 is 8.

Step 1: Identifying the numbers

Identify the two numbers that we need to add: 14 (a positive number) and -6 (a negative number).

Step 2: Understand the addition of positive and negative numbers

When adding a positive number and a negative number, it's like finding the difference between the two numbers. In this case, we'll subtract the absolute value of the negative number from the positive number.

Step 3: Subtract the absolute value of the negative number from the positive number

Subtract the absolute value of -6 (which is 6) from the positive number 14: \(14 - 6 = 8\)

Step 4: Write the final answer

The sum of \(14+(-6)\) is 8.

Key Concepts

Positive and Negative Numbers

Positive and negative numbers are fundamental concepts in mathematics. Positive numbers are those greater than zero and are usually written without a sign, like +14. Negative numbers, on the other hand, are less than zero and have a minus sign before them, like -6.

• Positive numbers: numbers to the right of zero on a number line.
• Negative numbers: numbers to the left of zero on a number line.
• Zero is neither positive nor negative and acts as the boundary between them.

Understanding how these numbers interact is essential. When combining them, they affect each other depending on their values and signs. Learning to manage them is a key step in mastering algebra and basic arithmetic.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It's always a non-negative value. For example, both +6 and -6 have an absolute value of 6.

• The absolute value function is denoted by vertical bars: \(|x|\).
• It simply removes the sign of the number, showing how far the number is from zero.
• This concept is crucial when comparing numbers, especially when handling negative numbers.

In the exercise, the absolute value of -6 is used to determine how much the negative number influences the positive number during addition.

Basic Arithmetic

Basic arithmetic is the foundation of mathematics involving simple operations such as addition, subtraction, multiplication, and division. These operations are essential for everyday calculations and problem solving.

• Addition: Combining values to get a total or sum.
• Subtraction: Taking away a value from another to find the difference.
• Multiplication: Repeated addition of a number by itself a specific number of times.
• Division: Splitting a number into equal parts or groups.

The current exercise primarily deals with addition involving both positive and negative numbers, requiring a good grasp of these basic operations.

Subtraction Method for Addition

The subtraction method for addition is a useful technique when adding a positive number and a negative number. Instead of summing directly, you subtract the absolute value of the negative number from the positive number. This method simplifies the process and is often more intuitive.

• Identify the numbers involved – in this case, +14 and -6.
• Find the absolute value of the negative number, which is 6.
• Subtract the absolute value from the positive number: 14 - 6.
• This brings you to the result, which is 8 in the example given.

This approach highlights the interaction between positive and negative numbers, focusing on their relative sizes and making such problems easier to solve.