Problem 27 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(x - 4\)

Step 1: Identify the Action

In the phrase 'four less than a number', the action that is being described is subtraction. The number four is being subtracted from another number.

Step 2: Identify the Involved Elements

The elements involved in the expression are the number four and another unidentified number, which we represent as \(x\).

Step 3: Write the Algebraic Expression

Considering the above, the phrase 'four less than a number' can be written in algebraic form as \(x - 4\), which means the number \(x\) minus four.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations, such as addition, subtraction, multiplication, and division. These expressions do not have an equality sign, unlike equations, and are used to represent mathematical ideas concisely. For example, when you hear a phrase like "four less than a number," it can be translated into an algebraic expression by identifying the numbers and operations involved.
In our example, we're dealing with the operation of subtraction. Here, the expression becomes "a number minus four" or algebraically represented as \(x - 4\). When tackling algebraic expressions, it is crucial to understand both the numbers involved and their relationship, as described by the arithmetic operations.
To effectively translate phrases into algebraic expressions, keep in mind:

• Identify the operation (e.g., addition, subtraction, etc.).
• Determine the numbers and variables involved.
• Organize them into a coherent expression.

Practicing these steps helps sharpen your ability to interpret and manage algebraic expressions.

Variables in Algebra

Variables are fundamental in algebra, representing unknown or changing numerical values. Think of them as placeholders that allow you to generalize mathematical expressions. They provide flexibility, enabling you to solve problems involving unknown numbers. Typically, letters such as \(x\), \(y\), or \(z\) are used as variables. For example, in the phrase "four less than a number," the unknown number can be represented as \(x\).
Using variables allows algebra to model real-life situations involving quantities that can vary. This abstraction makes it easier to solve a wide range of math problems.

• Variables can stand for any number or a specific number, as needed.
• They make it possible to generalize mathematical operations.
• They extend the ability to formulate and solve equations.

Once you're comfortable using variables, transforming real-world problems into simplified algebraic expressions becomes straightforward.

Basic Subtraction in Algebra

Subtraction is a basic arithmetic operation and a crucial component of algebraic expressions. In algebra, subtraction involves taking one quantity away from another, which is often expressed as \(a - b\), where \(a\) and \(b\) can be numbers, variables, or more complex expressions.
In the context of our example, "four less than a number," the phrase indicates that the number four is subtracted from another number represented by the variable \(x\). This is written as \(x - 4\).
Subtraction in algebra helps to:

• Modify quantities, showing how one value is reduced.
• Reorganize expressions for ease of solving equations.
• Simplify complex problems by breaking them down into parts.

Grasping the concept of subtraction within algebra allows for greater understanding and manipulation of quantities involved in mathematical expressions, aiding in solving equations and modeling real-world scenarios effectively.