Problem 106 - Elementary and Intermediate Algebra

Question

Translate each phrase to mathematical symbols. Let \(x\) represent the unknown number. a. 19 less than a number b. 19 is less than a number

Step-by-Step Solution

Answer

a) \(x - 19\); b) \(19 < x\)

Step 1: Understand the Phrase '19 less than a number'

The phrase '19 less than a number' means you are taking 19 away from the unknown number. If the unknown number is represented by \(x\), then '19 less than a number' translates to \(x - 19\).

Step 2: Translate '19 is less than a number'

'19 is less than a number' indicates an inequality. It means the number is greater than 19. Represent the number by \(x\), so the inequality is \(19 < x\).

Key Concepts

Inequalities

Inequalities are a way to express when one value is not equal to another and can be greater or less. In math, we use symbols like \(<\), \(>\), \(\leq\), and \(\geq\) to show these relationships.
Consider the phrase "19 is less than a number," which is telling us that compared to this number, 19 is smaller. Here, if the unknown number is represented by \(x\), the inequality can be expressed as \(19 < x\). This tells us that \(x\) is greater than 19.
Inequalities are crucial for solving real-world problems where precise equivalence is not always necessary. They help us understand a range of possible values that could make an equation true. For example, knowing that a temperature must be above a certain degree for a chemical reaction.

Expressions

Expressions in mathematics are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. They do not include an equality or inequality sign.
When we translate the phrase "19 less than a number," we use subtraction because we are removing 19 from the number. Representing the number again with \(x\), the expression becomes \(x - 19\).
Expressions are flexible and can be manipulated through algebraic operations to simplify them or to solve equations. Understanding expressions is key to writing solutions and making calculations.

Mathematical Representation

Mathematical representation involves converting real-world problems or statements into mathematical symbols and expressions. This transformation helps solve problems efficiently.
By representing words like "19 less than a number" with the expression \(x - 19\), and "19 is less than a number" with the inequality \(19 < x\), we organize information clearly.
These representations allow us to use algebraic rules to evaluate and find solutions. Turning words into math symbols is essential for solving many types of math problems, as it translates language into the universal language of mathematics.