Problem 65 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression and simplify if possible. Let \(x\) represent the unknown number. Twice a number, decreased by four

Step-by-Step Solution

Answer

The algebraic expression is \(2x - 4\).

Step 1: Identify Key Words

First, read the phrase carefully: 'Twice a number, decreased by four.' Identify what each part means in algebraic terms. 'Twice a number' suggests multiplication by 2, and 'decreased by four' means subtracting 4.

Step 2: Translate Words to Algebraic Expression

Using the understanding from the previous step, translate the phrase into an algebraic expression. 'Twice a number' would be written as \(2x\), and 'decreased by four' tells us to subtract 4 from this product.

Step 3: Write the Algebraic Expression

Combine the components of the phrase into a single algebraic expression: \(2x - 4\). This represents 'twice a number, decreased by four.'

Step 4: Check for Simplification

The expression \(2x - 4\) cannot be simplified further because it is already in its simplest form, with no like terms to combine or common factors to factor out.

Key Concepts

Algebraic Translation

Algebraic translation is the process of converting a verbal or written phrase into an algebraic expression. This skill is essential in mathematics, especially when dealing with problems where information is provided in a textual format. Understanding how to translate words into mathematical symbols allows you to solve real-world problems more effectively.

To translate verbal phrases into algebraic expressions, identify key terms that indicate mathematical operations. For example:

• 'Twice a number' indicates that a number is multiplied by 2, resulting in the expression \(2x\).
• 'Decreased by' implies subtraction. So, 'decreased by four' suggests subtracting 4.

Breaking down the sentence "Twice a number, decreased by four," we first translate "twice a number" into \(2x\), representing the multiplication of the unknown number by 2. Then, we subtract 4 to form the complete expression \(2x - 4\). Translating correctly is vital because any errors at this stage can lead to incorrect solutions in later steps.

Simplification

Simplification in algebra involves reducing an algebraic expression to its most concise form, where no further combining of like terms or factoring is possible. This makes the expression easier to work with, especially when performing additional calculations or when comparing different expressions.

In our example, once the expression \(2x - 4\) is formed, we must check if it can be simplified further. Simplification involves:

• Combining like terms: Identifying and summing up terms that have the same variable parts.
• Factoring: Factoring out common factors from terms to simplify expressions, whenever applicable.

For \(2x - 4\), combining like terms or factoring are not applicable since there are no like terms to combine and no common factor outside the numeric coefficient. Hence, \(2x - 4\) remains in its simplest form. Recognizing when an expression has been fully simplified is key to preventing unnecessary or erroneous manipulations.

Mathematical Operations

Mathematical operations are the foundation of manipulating algebraic expressions. In this context, they include the basic operations: addition, subtraction, multiplication, and division.

In translating the phrase "Twice a number, decreased by four," we employed:

• Multiplication to find "twice a number," which gave us \(2x\), as multiplying an unknown number \(x\) by 2 results in this expression.
• Subtraction for "decreased by four," which is straightforward as reducing the result of the first operation by 4 directly leads us to the expression \(2x - 4\).

Understanding these basic operations and how they apply to algebraic expressions is essential, as they form the building blocks for solving complex algebraic equations. Knowing when and how to apply these operations correctly ensures accurate translation from language to mathematics and allows for precise simplification and solution derivation.