Problem 84 - Introductory and Intermediate Algebra for College Students 4th

Question

Write as an algebraic expression in which \(x\) represents the number: the quotient of 9 and a number, decreased by 4 times the number. (Section 1.1, Example 3)

Step-by-Step Solution

Answer

The algebraic expression that represents the given sentence is \(\frac{9}{x} - 4x\).

Step 1: Identify and represent the term 'quotient of 9 and a number'

The term 'quotient of 9 and a number' can be represented mathematically as \(\frac{9}{x}\), where \(x\) represents the unknown number.

Step 2: Identify and represent the term '4 times the number'

The term '4 times the number' can be represented as \(4x\), where \(x\) is the unknown number.

Step 3: Combine the parts

We have to represent the whole sentence as an algebraic expression. The sentence indicates that the quotient is decreased by 4 times the number. In mathematical terms, this can be represented as \(\frac{9}{x} - 4x\).

Key Concepts

Quotient

A quotient is the result you get when you divide one number by another. In mathematical terms, if you have two numbers, say 9 and x, their quotient is written as \(\frac{9}{x}\). Here, the symbol \(\div\) or fraction line means 'divide'. So \(\frac{9}{x}\) literally translates to "9 divided by some number x."
This concept is one of the building blocks in algebraic expressions and helps in representing situations involving division.
Understanding fractions and quotients helps you solve problems related to splitting or distributing items evenly.

Unknown Number

In algebra, 'x' is often used to represent an unknown number. This simply means that 'x' stands for a value that is not known yet. It's like a placeholder that you can replace once you find out what the actual number is.
Using an unknown number allows us to write flexible equations that can be solved to find out this mystery number.
For example, if you use 'x' in an expression like \(4x\), it stands for "4 times an unknown number". This is very useful in equations where you have to solve for 'x' to find its actual value.

Mathematical Representation

Mathematical representation involves expressing words or phrases as numbers and symbols. It helps in converting real-life problems into a form that you can solve using mathematical methods.
To mathematically represent 'the quotient of 9 and a number, decreased by 4 times the number', we break it down into parts:

• "quotient of 9 and a number" is \(\frac{9}{x}\)
• "4 times the number" is \(4x\)

Then, we combine them into one expression: \(\frac{9}{x} - 4x\). This representation gives us a clear mathematical sentence that describes the problem.

Subtracting Expressions

Subtracting expressions is about finding the difference between two algebraic expressions. In this case, it involves the expression "\(\frac{9}{x}\) - 4x".
Here, you're subtracting \(4x\) from \(\frac{9}{x}\). Subtraction does exactly what it sounds like – it takes part of the value away from another.
To perform the subtraction, just imagine what it would mean to start with one term and take away the other. This operation helps in simplifying complex equations and finding solutions.