Problem 66 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ 4+2[-7-(3-9)] $$

Step-by-Step Solution

Answer

The expression evaluates to 2.

Step 1: Solve Inside the Parentheses

Begin by addressing the innermost set of parentheses: \(3 - 9\). Calculate this as: \(3 - 9 = -6\). Now the expression becomes \[4 + 2[-7-(-6)]\].

Step 2: Handle the Negative Sign

Evaluate the expression inside the brackets: \(-7 - (-6)\). This is equivalent to \(-7 + 6\), because subtracting a negative is like adding. Thus, \(-7 + 6 = -1\). The expression now simplifies to \[4 + 2[-1]\].

Step 3: Multiply inside the Brackets

Multiply the number outside the brackets by the number inside: \(2[-1]\). Calculate \(2 \times -1 = -2\). Now the expression is reduced to \[4 - 2\].

Step 4: Final Addition

Perform the final addition or subtraction: \(4 - 2\). This simplifies to \(2\). The entire expression evaluates to \(2\).

Key Concepts

Order of Operations

The order of operations is a fundamental principle in algebra and ensures consistent results when evaluating mathematical expressions. It can be remembered by the acronym PEMDAS, which stands for:

• P: Parentheses
• E: Exponents (powers and roots)
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

Each type of operation must be completed in this order, starting from the top. For instance, operations within parentheses have to be completed before you move on to exponents, and so on.
When expressions have multiple operations at the same level (eg., addition and subtraction, or multiplication and division), they are performed from left to right.
In our example, the expression is first simplified inside the parentheses. After that, multiplication is performed before addition and subtraction. This careful sequence ensures an accurate result every time.

Parentheses

Parentheses are used to indicate that the operations enclosed within them have a higher priority. They change the standard order of operations by ensuring what’s inside is calculated first.
In the given example, the expression initially within the parentheses is \(3 - 9\). Even if there are other calculations outside, whatever happens inside the parentheses must be completed first.
Once we've handled that, the parentheses can be "removed" by substituting in the result of the inner calculation into the larger expression, allowing us to continue simplifying using the rules of order of operations.

Negative Numbers

Negative numbers add a layer of complexity to arithmetic operations. Understanding their behavior is crucial, especially when they are involved in subtraction or multiplication.
In the exercise, the challenge comes when dealing with the expression \(-7 - (-6)\). Here, two negative signs appear consecutively: subtracting a negative is equivalent to adding the positive counterpart. So, \(-7 - (-6)\) simplifies as \(-7 + 6\), resulting in \(-1\).
Later in the process, we encounter multiplication of negative numbers. Multiplying \(2\) and \(-1\) involves taking one positive number and one negative, giving us \(-2\).
Handling these operations carefully ensures the correct simplification of the entire expression.