Problem 15 - Elementary Algebra Within Reach

Question

In Exercises \(11-22\), translate the verbal phrase into an algebraic expression. $$ \text { Six less than } g $$

Step-by-Step Solution

Answer

The algebraic expression is \( g - 6 \)

Step 1: Identify the quantities and operations

In the given statement, the quantities mentioned are 'six' and the variable 'g'. The operation mentioned is 'less than', which implies subtraction. In English, 'six less than g' means that we are removing six units from the quantity 'g'.

Step 2: Understand the structure of the phrase

In English mathematical statements, the phrase 'less than' suggests that the quantity you are subtracting (i.e., 'six' in this case) comes after the quantity from which it is subtracted (i.e., 'g'). This is actually the reverse of how it's written in algebra.

Step 3: Translate into an algebraic expression

Given the understanding from steps 1 and 2, the correct translation of 'Six less than g' into an algebraic statement would be 'g - 6'.

Key Concepts

Translation of Verbal Phrases into Algebraic Expressions

Translating verbal phrases into algebraic expressions is a critical skill in algebra. It involves turning written words into mathematical equations or expressions. This requires understanding the specific keywords and operations embedded in the phrase. Consider the phrase "six less than \( g \)". Here we need to:

• Identify the numbers and variables involved, like 'six' and the variable 'g'.
• Understand the operation indicated by words like "less than", which means subtraction.

Generally, mathematical sentences are structured differently than everyday language. In this expression, "less than" signals not only subtraction but also that the number 'six' should come after the variable 'g' in the expression. Thus, we write this as \( g - 6 \). This reverse order from English to algebra is a common area to watch closely while translating.

Subtraction in Algebra

Subtraction in algebra works similarly to subtraction in basic arithmetic, but with the inclusion of variables. It is essential to understand how to rearrange expressions based on linguistic cues.
When given "six less than \( g \)", subtraction is used to decrease the value of \( g \) by six. This operation is directly translated to \( g - 6 \) in algebraic form. A helpful way to remember this is:

• "Less than" implies that something is being taken away from another quantity.
• Always place the number being subtracted after the variable or quantity in the expression.

This structure is crucial. Misplacing the numbers or variables can lead to incorrect results.
Remember, algebra is about maintaining the relationships and order that govern mathematical logic. Mastering how to interpret these verbal phrases correctly ensures accurate mathematical translations.

Mathematical Statements

A mathematical statement is a clear, precise declaration about a mathematical situation or relationship, formed by connecting math expressions.
It embodies both functions and equations and can describe quantities, relationships, or operations. When dealing with phrases such as "six less than \( g \)", you are asked to express a particular relationship using symbols and numbers.

• A statement like \( g - 6 \) is an expression showing a relationship between 'g' and 'six'.
• Mathematical statements must be unambiguous and correctly represent the intended operation.

The power of mathematical statements comes from their ability to universally convey meaning without language barriers. When created correctly, they can be universally understood, making math a global language.
In summary, becoming proficient in translating verbal phrases into algebraic expressions enriches your ability to formulate precise mathematical statements that accurately represent real-world relationships.