Problem 23 - Algebra 1

Question

Write the verbal sentence as an equation or an inequality. Nine is greater than three times a number \(s\).

Step-by-Step Solution

Answer

The mathematical inequality for the verbal sentence 'Nine is greater than three times a number \(s\)' is \(9 > 3s\)

Step 1: Identify and Understand the Keywords

In the verbal sentence, 'Nine is greater than three times a number \(s\)' the key terms are: 'greater than' which is an inequality, and 'times' denoting multiplication. The number 'nine' and the variable '\(s\)' are the quantities being compared.

Step 2: Translate the Verbal Sentence to a Mathematical Inequality

The verbal sentence can be translated into a mathematical inequality as follows: The number 'nine' corresponds to 9. The phrase 'is greater than' corresponds to the inequality '>'. 'Three times a number \(s\)' indicates 3 multiplied by \(s\) or \(3s\). Hence the translated inequality would be \(9 > 3s\).

Key Concepts

Inequality

An inequality is a mathematical statement that compares two values, expressions, or quantities, indicating that one is larger or smaller than the other. Inequalities are fundamental in expressing relationships where quantities are not equal and come in various forms, such as '<' (less than), '>' (greater than), '(\leq)' (less than or equal to), and '(\geq)' (greater than or equal to).In our exercise, the inequality is used to compare the value nine with three times a number (\(s\)). The phrase 'greater than' indicates that nine is on the larger side of the comparison, pointing us toward the '>' symbol in mathematical notation. Inequalities like this are crucial for modeling situations that deal with ranges of values or conditions, often appearing in real-world problems such as those involving financial budgeting, statistics, or optimization.

Mathematical Translation

Mathematical translation involves converting words and phrases from a verbal or written description into a precise mathematical form—usually equations or inequalities. This skill is essential in interpreting and solving math problems that are posed in natural language.In our example, the verbal sentence 'Nine is greater than three times a number \(s\)' is translated into an inequality. Each part of the sentence correlates to a mathematical component: 'nine' becomes '9', 'greater than' translates to '>', and 'three times a number \(s\)' is represented as '3\(s\)'. The translation becomes clearer by identifying keywords and understanding their mathematical counterparts, turning a structured sentence into an algebraic inequality '9 > 3\(s\)'.

Variable Representation

In algebra, a variable is a symbol used to represent an unknown or changeable quantity. Here, the variable \(s\) stands for an unspecified number—think of it as a placeholder for any real number.Variables are powerful because they are the building blocks that allow us to create general statements like equations or inequalities that can be true for many different values. They also help in formulating and solving problems mathematically. The sentence from our exercise introduces \(s\) as a number that we do not yet know, but the inequality gives us a relationship between \(s\) and the known quantity nine, allowing us to determine the range of possible values \(s\) can take.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations (such as addition, subtraction, multiplication, and division) that represent a particular quantity or set of quantities.In our task, the expression 'three times a number \(s\)' is an algebraic expression that involves the variable \(s\) and the arithmetic operation of multiplication. To interpret it mathematically, we understand that 'times' indicates multiplication, so 'three times \(s\)' is expressed as '3\(s\)'—an algebraic expression signifying 3 multiplied by the value of \(s\). Algebraic expressions are tools that, when combined with equality or inequality signs, become algebraic equations or inequalities representing a wide range of problems and their solutions.