Problem 36 - Beginning and Intermediate Algebra

Question

Find each sum. $$ 6+[12+(-3)] $$

Step-by-Step Solution

Answer

15

Step 1 - Identify the Expression Inside the Brackets

First, evaluate the expression inside the brackets. Here, we need to add 12 and -3.

Step 2 - Perform Addition Inside the Brackets

Calculate the sum of 12 and -3: 12 + (-3) = 9

Step 3 - Add the Result to the Outer Number

After calculating the sum inside the brackets, we have 9. Now add this to the outer number, 6: 6 + 9 = 15

Key Concepts

Order of Operations

Understanding the order of operations is crucial in solving math problems accurately. The order of operations refers to the rules that dictate the sequence in which operations should be performed. In mathematics, this sequence can be remembered by the acronym PEMDAS, which stands for:

• P - Parentheses
• E - Exponents
• M - Multiplication
• D - Division
• A - Addition
• S - Subtraction

Following PEMDAS ensures we get the correct answer. Failing to follow it can lead to incorrect results. For example, in our original exercise, we followed the order of operations to first evaluate the expression inside the brackets before performing addition.

Bracket Evaluation

Brackets, also known as parentheses, are used in math to group parts of an expression that should be treated as a single unit. Evaluating the content within the brackets first is essential, as per the order of operations.
In our specific problem, we needed to evaluate the expression inside the brackets \(12 + (-3)\) before anything else. Here’s the step-by-step breakdown:

• Identify the expression inside the brackets: \(12 + (-3)\)
• Perform the arithmetic operation inside the brackets: 12 and -3
• Calculate the sum: \(12 + (-3) = 9\)

Once we have the result from inside the brackets, we can proceed to the next part of the expression.

Integer Arithmetic

Working with integers involves understanding both positive and negative numbers. Integer arithmetic includes operations like addition, subtraction, multiplication, and division on these whole numbers. Here, we focus on addition involving a positive and a negative integer.
In the step \(12 + (-3)\):

• The number 12 is a positive integer.
• The number \(-3\) is a negative integer.
• When adding a positive integer and a negative integer, we can think of it as a subtraction. Essentially, we subtract the absolute value of the negative number from the positive number: \(12 - 3 = 9\)

Understanding these rules helps in accurately performing arithmetic with integers.

Addition

Addition is one of the fundamental operations in mathematics. It involves combining two or more numbers to get their total sum. In the given exercise, after solving the expression inside the brackets, we performed an addition.
Here’s the step-by-step:

• First, solve the inner expression: \(12 + (-3) = 9\)
• Then, add this result to the outer number: \(6 + 9\)
The result of \(6 + 9\) is 15. Therefore, the final answer is 15.
Addition is straightforward when dealing with positive numbers. However, it becomes slightly complex with negative numbers, which is why understanding integer arithmetic is essential. Combining these skills ensures accurate calculations in any math problem.