Problem 18 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-19\).

Step 1: Identify and Prioritize Operations

The expression given is \(9-(3+11) \times 2\). According to the order of operations (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) - PEMDAS), we first need to calculate the expression inside the parentheses.

Step 2: Simplify Inside the Parentheses

Within the parentheses, we have \(3+11\). Solving this, we get: \(3+11=14\). Now the expression becomes \(9-14 \times 2\).

Step 3: Perform Multiplication

Next, according to the order of operations, we perform the multiplication \(14 \times 2\), which equals \(28\). Now our expression is simplified to \(9-28\).

Step 4: Perform Subtraction

Finally, we perform the subtraction: \(9-28\). The result of this operation is \(-19\).

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions involves reducing them to their simplest form. This helps in understanding and solving equations more easily. Simplification involves eliminating unnecessary parts of the equation, making it shorter and easier to work with. To simplify, follow these steps:

• Look for operations within parentheses and handle them first.
• Combine like terms to reduce the number of terms in the expression.
• Solve any arithmetic calculations such as multiplication or division.

Although the order in which you simplify certain operations may vary slightly, the goal is always to arrive at a form that is as straightforward as possible. This process helps in clarifying what the expression represents and prepares it for further mathematical operations.

PEMDAS

PEMDAS is an acronym used to remember the order of operations in mathematics:

• P: Parentheses first
• E: Exponents (powers and roots, etc.)
• MD: Multiplication and Division (left-to-right)
• AS: Addition and Subtraction (left-to-right)

When simplifying expressions like the one in the exercise, starting with operations inside parentheses is crucial. This prevents misinterpretations in the sequence of operations. For example, without applying PEMDAS, the expression \[ 9-(3+11) \times 2 \] could be mistakenly simplified incorrectly by someone performing operations out-of-sequence. Following PEMDAS ensures that multiplication is correctly prioritized after handling any additions inside brackets.

Algebraic Operations

Algebraic operations include different mathematical tasks such as addition, subtraction, multiplication, and division applied to variables and numbers within expressions. Understanding these operations is fundamental to solving algebra problems effectively.In the given example, \[ 9-(3+11) \times 2 \]start by adding the numbers within the parentheses first. This involves basic addition, a core algebraic operation, resulting in 14. Then, proceed to multiplication, another key operation, between 14 and 2. Finally, perform subtraction by taking 28 away from 9. Each of these steps demonstrates an algebraic operation, showing how they combine to simplify the expression fully. Through these operations, expressions are systematically reduced and solved.