Problem 18 - College Algebra

Question

For the following exercises, simplify the given expression. $$ 9-(3+11) \cdot 2 $$

Step-by-Step Solution

Answer

The simplified expression is \(-19\).

Step 1: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses. The expression is \(3 + 11\). Compute this by adding the two numbers: \(3 + 11 = 14\). Thus, the expression becomes \(9 - 14 \cdot 2\).

Step 2: Perform Multiplication

Now proceed by multiplying \(14\) and \(2\). Calculate this as \(14 \cdot 2 = 28\). Thus, the expression reduces to \(9 - 28\).

Step 3: Perform the Subtraction

Next, perform the subtraction \(9 - 28\). Since \(28\) is larger than \(9\), the result will be negative: \(9 - 28 = -19\).

Key Concepts

Understanding the Order of Operations

In mathematics, the order of operations establishes the rules that dictate the sequence in which operations are performed in an arithmetic expression. This is crucial as it ensures consistency and accuracy in calculations. The most common method to remember this order is the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). By following these steps, you can tackle complex expressions efficiently.

The order of operations directs us to first solve any calculations inside parentheses. In the given problem, we begin with the expression inside parentheses, i.e., \(3 + 11\). Once parentheses are resolved, the next step is to address any multiplication or division, which corresponds to \(14 \cdot 2\) in our problem. Finally, subtraction is performed as showcased by \(9 - 28\).

• Always start with operations inside parentheses.
• Handle exponents next, if present.
• Proceed with multiplication and division from left to right.
• Complete the calculation with addition and subtraction from left to right.

Grasping Arithmetic Expressions

Arithmetic expressions consist of numbers, operators, and sometimes parentheses. They are the building blocks of algebra and essential in simplifying any mathematical problem. The primary goal of working with these expressions is to reach the simplest form or find the result of the expression.

In the problem \(9 - (3 + 11) \cdot 2\), the arithmetic expression involves basic operations such as addition, multiplication, and subtraction. Each operation affects the overall result and, when combined, shapes the path to simplifying the expression. By understanding the correct order and technique, interpreting even complex expressions becomes manageable.

• Addition combines values together.
• Multiplication repeats addition a specific number of times.
• Subtraction denotes a decrease by a value.

Following Basic Algebra Steps

Algebra often involves simplifying expressions and solving for variables, but even without variables, following algebra steps is essential in finding solutions in a structured manner. Let's break down the process into simple steps using the provided problem as a guide.

### Step-by-Step Breakdown:
- **Evaluate Parentheses**: Initial evaluation begins with parentheses. Here, \((3 + 11)\) simplifies to \(14\).
- **Apply Multiplication**: Next, multiplication \(14 \cdot 2\) is computed, resulting in \(28\).
- **Subtract the Result**: Finally, perform the subtraction \(9 - 28\), which gives \(-19\).

This structured method allows for systematically approaching any expression. When faced with similar problems, always apply these fundamental steps.

• Start with elements in parentheses.
• Apply operations across factors outside parentheses.
• Complete the expression with addition or subtraction.