Problem 60 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. $$[8+(-2)]+[5+(-7)]$$

Step-by-Step Solution

Answer

The simplified expression is 4.

Step 1: Simplify Inside the First Bracket

We begin by simplifying the expression inside the first pair of brackets: \(8 + (-2)\). Add \(-2\) to \(8\): \(8 - 2 = 6\).

Step 2: Simplify Inside the Second Bracket

Next, simplify the expression inside the second pair of brackets: \(5 + (-7)\). Add \(-7\) to \(5\): \(5 - 7 = -2\).

Step 3: Combine the Results

Now, combine the results from Steps 1 and 2. You add the results from both brackets together: \(6 + (-2) = 6 - 2 = 4\).

Key Concepts

Simplifying Expressions

When simplifying expressions, the goal is to reduce them to their simplest form. This involves performing operations in an expression based on the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
To simplify an expression, you should:

• First, deal with any operations inside parentheses or brackets. This means solving any inner expressions first.
• Next, consider exponents (if there are any) after simplifying what's inside the brackets.
• Finally, perform addition or subtraction, ensuring each step you take simplifies the expression further.

By following these steps, expressions become easier to manage and calculate. Simplification makes complex problems simpler and often provides a clearer path to the correct answer.

Integer Addition

Adding integers is an essential part of simplifying expressions. When working with integers—whole numbers that can be positive, negative, or zero—you should consider both the number and its sign.

• Add two positive numbers by simply combining them.
• For two negative numbers, add them together and retain the negative sign, as the overall value will still be negative.
• To add a positive and a negative number, subtract the smaller magnitude from the larger magnitude. The sign of the larger magnitude remains with the result.

For example, in the original problem, we added \(8 + (-2)\) to get 6, because subtracting 2 from 8 leaves us with 6. Understanding these rules ensures you handle integer addition correctly in various math contexts.

Brackets in Expressions

Brackets—or parentheses—serve a crucial purpose in math expressions. They dictate which parts of an expression should be solved first, in adherence to the order of operations.
Here's how brackets function in simplifying:

• Brackets can serve to group parts of an expression that need solving before proceeding with the rest of the operations.
• Perform calculations inside the brackets before doing anything outside them to ensure proper simplification.
• If there are multiple levels of brackets, solve the innermost brackets first.

In the example provided, each part of the bracketed expression was simplified separately before combining the results. By correctly addressing the expressions inside brackets, the problem becomes manageable and less prone to error.