Problem 91 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ -(6-8)-(2-4) $$

Step-by-Step Solution

Answer

0

Step 1 - Simplify inside the parentheses

First, simplify the expressions inside each pair of parentheses. For the first set of parentheses, calculate: 6 - 8 = -2 For the second set of parentheses, calculate: 2 - 4 = -2

Step 2 - Apply the negative sign outside the parentheses

Next, apply the negative signs outside the parentheses to the simplified results: -(-2) - (-2)

Step 3 - Simplify the negative signs

Simplify the expression by removing double negatives: -(-2) = 2 -(-2) = 2 Therefore, the expression simplifies to: 2 - 2

Step 4 - Combine the final terms

Finally, combine the simplified terms: 2 - 2 = 0

Key Concepts

Arithmetic Operations

Arithmetic operations are basic mathematical operations that include addition, subtraction, multiplication, and division. These operations are essential in simplifying algebraic expressions and solving equations.
In the given exercise - dash or minus (-) denotes subtraction,
Here’s a simple rundown on arithmetic operations used in this example:

• Addition: Combining two quantities. For example, 3 + 2 = 5
• Subtraction: Taking one quantity away from another. For example, 6 - 4 = 2
• Multiplication: Repeated addition of a number. For example, 3 * 2 = 6
• Division: Splitting a number into equal parts. For example, 8 ÷ 2 = 4

Understanding these basic operations is crucial for handling more complex algebraic expressions.

Negative Numbers

Negative numbers are those less than zero, represented with a minus sign (-). In algebra, working with negative numbers is common and understanding how to manipulate them is critical.

• A negative number plus another negative number results in a more negative value. For example, -3 + (-2) = -5.
• A negative number minus another negative number results in a less negative value. For example, -5 - (-2) = -3.
• Multiplying two negative numbers results in a positive value. For example, -3 * -2 = 6.
• Dividing two negative numbers also results in a positive value. For example, -6 ÷ -2 = 3.

In this exercise, we simplify inside the parentheses first. For example, 6 - 8 results in -2 and 2 - 4 also results in -2. Understanding negative numbers helps us to correctly apply the subsequent steps in the simplification process.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which computations should be performed. Following this correctly assures that everyone gets the same result for a given mathematical expression.
A commonly used acronym for remembering the order of operations is PEMDAS which stands for:

• P: Parentheses – Simplify expressions inside parentheses first
• E: Exponents – Then resolve exponents (powers and roots)
• M/D: Multiplication and Division – Perform multiplication and division from left to right
• A/S: Addition and Subtraction – Last, carry out addition and subtraction from left to right

In the given exercise, following PEMDAS correctly means simplifying the expressions inside the parentheses first, and then applying the arithmetic operations as per the rule.

Parentheses Simplification

Parentheses are used in mathematical expressions to indicate that the operations enclosed should be performed first. Simplifying within the parentheses is the first step in following the order of operations.
In the provided example, we begin with simplifying the expressions inside the parentheses:

• 6 - 8 = -2
• 2 - 4 = -2

After simplifying inside the parentheses, the expression becomes -(-2) - (-2). Rewriting these expressions helps to handle any negative signs associated with the parentheses. Applying the rules for working with negative numbers, we know that -(-2) simplifies to 2 and -(-2) again simplifies to 2. Finally, we subtract 2 - 2, which results in 0. Simplifying parentheses correctly ensures the accurate simplification or solution of the entire expression.