Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. four more than a number

Step-by-Step Solution

Answer

The algebraic expression for the English phrase `four more than a number` is \(x + 4\).

Step 1: Identify the Variable

The problem states that the variable \(x\) represents the number.

Step 2: Understand the Phrase

`Four more than a number` indicates that 4 is being added to a number.

Step 3: Write the Algebraic Expression

Combine the variable and the number in an addition operation. This gives us \(x + 4\).

Key Concepts

Variables in Algebra

In algebra, variables are symbols used to represent numbers that may vary, typically denoted by letters such as x, y, or z. These variables are the cornerstone of algebraic expressions, equations, and functions, as they allow students and mathematicians to describe and work with quantities that aren't fixed.

When dealing with variables, it's crucial to comprehend that they can take on any value. In the context of our exercise, the variable x represents an unknown number that we're working with. One of the most common mistakes is not clearly defining your variable at the outset, but as demonstrated in the exercise, stating that x represents 'the number' ensures clarity and a firm understanding as you progress through algebraic problems.

Translating English to Algebra

One of the crucial skills in algebra is translating English phrases into algebraic expressions. This involves recognizing key phrases and understanding their mathematical significance. For example, the phrase 'four more than a number' necessitates an addition operation in algebra.

Common Phrases and Their Algebraic Equivalents

• 'More than' usually signifies addition.
• 'Less than' typically means subtraction.
• 'Times', 'multiplied by', or 'product of' indicate multiplication.
• 'Divided by' or 'ratio of' suggest division.

By breaking down the phrase into smaller parts and identifying these keywords, one can effectively translate everyday language into the language of algebra, making the process of solving problems much more intuitive.

Basic Algebra Operations

Algebra relies heavily on four fundamental operations: addition, subtraction, multiplication, and division. These operations allow us to manipulate algebraic expressions to simplify or solve them.

In the solution of our exercise, we focus on addition, denoted by the plus sign (+). As with arithmetic, in algebra, addition combines quantities. Subtraction, multiplication, and division similarly follow their arithmetic counterparts but can look slightly different when variables are involved.

Understanding how to properly perform these operations with variables is essential. Remember, when combining like terms (terms with the same variables raised to the same power), you add or subtract the coefficients only, while multiplication and division can result in power changes or rational expressions.