Problem 108 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ 23-2(4-6) $$

Step-by-Step Solution

Answer

27

Step 1 - Understand the expression

The expression given is \[23 - 2(4 - 6)\]

Step 2 - Simplify inside the parentheses

First, solve the operation inside the parentheses. \[4 - 6 = -2\].

Step 3 - Substitute the result back into the expression

Replace the parentheses in the original expression with the simplified value: \[23 - 2(-2)\]

Step 4 - Apply the multiplication

Next, multiply the number outside the parentheses by the result inside: \[2 \times -2 = -4\]. So, the expression becomes \[23 - (-4)\]

Step 5 - Simplify the final expression

Now simplify the expression by recognizing that subtracting a negative is the same as adding: \[23 + 4 = 27\]

Key Concepts

Understanding the Order of Operations

In mathematics, the order of operations is a rule we follow to ensure we get the correct result. These rules help us decide which operations to perform first in an expression. The commonly used acronym to remember this order is PEMDAS:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

Always perform operations inside parentheses first, then exponents (like squares or square roots), followed by multiplication and division from left to right, and finally addition and subtraction from left to right. This ensures that the solution follows a clear and consistent method.

Using Parentheses Effectively

Parentheses ( ) are used in algebra to group parts of an expression that should be evaluated first. For example, in the expression \[23 - 2(4 - 6)\], the part inside the parentheses \(4 - 6\) should be solved first.

Step-by-step, this looks as follows:
- Solve the expression inside the parentheses: \4 - 6 = -2\.
- Replace the original parentheses with the result: \23 - 2(-2)\.

Paying close attention to parentheses ensures we don't mix up the order in which operations are performed, leading to correct results every time.

Working with Negative Numbers

Negative numbers can seem tricky, but they follow the same rules as positive numbers with some signs flipped. Here are a few key rules to remember:

• Subtracting a positive number is the same as adding a negative: \ 23 - 2 = 23 + (-2) \.


• Subtracting a negative number is the same as adding the positive: \ 23 - (-2) = 23 + 2 \.


• Multiplying two negative numbers gives a positive result, while multiplying a positive and a negative number gives a negative result: \[2 \times -2 = -4\]

Understanding negative numbers can help improve overall comfort with algebraic operations, making it easier to solve complex expressions.

Basic Arithmetic Operations

Understanding basic arithmetic operations like addition, subtraction, multiplication, and division are essential in algebra. Here’s a quick breakdown:

• Addition: Combine two numbers to get their total. Example: \5 + 3 = 8.\
• Subtraction: Determine the difference between two numbers. Example: \10 - 7 = 3\. When subtracting negative numbers, add them instead: \[23 - (-4) = 23 + 4 = 27\]
• Multiplication: Find the product of two numbers. Example: \[2 \times 3 = 6\]
• Division: Determine how many times one number is contained in another. Example: \[8 \div\ 2 = 4\]

Returning to our earlier example: after replacing the parentheses value, you multiply: \[2 \times -2 = -4\] and finally simplify: \[23 - (-4) = 23 + 4 = 27\]. Mastering these operations is key to solving algebraic expressions efficiently.