Problem 27 - 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 less than a number

Step-by-Step Solution

Answer

The algebraic expression for 'four less than a number' is \(x - 4\).

Step 1: Identify the Variable

The problem statement provides that the variable 'x' represents the number. This is the variable that will be used in the expression.

Step 2: Determine the Operation

The word 'less' indicates subtraction. However, it's important to note that 'less than' signifies that subtraction happens in reverse order. So, 'four less than a number' means 'the number minus 4'.

Step 3: Form the Expression

Finally, substitute 'the number' with 'x' and form the expression accordingly. In this context, the phrase translates to 'x - 4', which represents 'four less than a number'.

Key Concepts

Variables in Algebra

In algebra, variables are symbols used to represent unknown or changeable quantities. They allow us to express general mathematical ideas without specifying exact numbers. Usually, letters from the alphabet, like \(x\), \(y\), or \(z\), are used as variables.
For example, in the phrase 'four less than a number', 'a number' is an unknown value that we can represent with the variable \(x\).

• A variable can represent any value which is not yet known or needs to be found.
• Using variables instead of specific numbers helps create expressions that can be altered by changing the variable's value.

This flexibility is what makes algebra such a powerful tool in solving mathematical problems.

Subtraction Operation

Subtraction is one of the basic operations in mathematics. It is used when we need to find the difference between numbers or quantities. Often, it helps answer the question, 'How much is left after taking some away?'. In algebra, subtraction can also involve expressions with variables.
When you see the word 'less' in a phrase like 'four less than a number', it means you will be subtracting something. But remember, 'less than' implies that subtraction happens in a specific order.

• 'Four less than a number' can be visualized as starting with an unknown number (represented by \(x\)) and subtracting four from it.
• So, the expression becomes \(x - 4\).

This subtlety in language is crucial for forming correct expressions.

Writing Expressions

Writing expressions involves translating words into mathematical language using symbols and numbers. This skill bridges the gap between verbal descriptions of problems and their mathematical representations.

• Identify key terms: Look for words like 'more than', 'less than', or 'increased by', which indicate specific mathematical operations.
• Choose a variable to represent unknowns: As you've seen, 'a number' can be represented by \(x\).
• Order matters: The arrangement of numbers and variables in the expression must align with the intended meaning of phrases, such as ensuring that for 'four less than a number', the expression is written as \(x - 4\).

Through practice, writing expressions becomes more intuitive, turning words into equations and paving the way for their solution.