Problem 93 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The quotient of \(-2\) and a number, subtracted from the quotient of \(-5\) and the number.

Step-by-Step Solution

Answer

The algebraic expression representing the given English phrase is \(-\frac{3}{x}\).

Step 1: Write the initial phrases as algebraic expressions

Start by translating the English phrases into algebraic format. The term 'the quotient of -2 and a number' becomes \(-\frac{2}{x}\). The next phrase 'the quotient of -5 and the number' translates to \(-\frac{5}{x}\). 'Subtracted from' is the operation between these two.

Step 2: Assemble the full expression

The full expression incorporates both parts, with 'subtracted from' implying subtraction: \(-\frac{5}{x} - -\frac{2}{x}\). Because we are subtracting a negative fraction, this becomes addition.

Step 3: Simplify the expression

Simplify the expression to its simplest form: \(-\frac{5}{x} + \frac{2}{x}\) simplifies to \(-\frac{3}{x}\).

Key Concepts

Quotient

In mathematical terms, a quotient refers to the result of a division problem. This is foundational in algebra, where we deal with expressions involving numbers. Understanding how to handle quotients is crucial to manipulating and simplifying algebraic expressions.
This exercise involved the terms "the quotient of -2 and a number" and "the quotient of -5 and the number." The phrase "quotient of -2 and a number" indicates we are dividing -2 by the variable, usually represented by \( x \). Therefore, it is written as \( -\frac{2}{x} \). Similarly, "quotient of -5 and the number" becomes \( -\frac{5}{x} \).
The concept of a quotient is vital when simplifying expressions, especially when subtracting or adding such terms in algebra.

Simplifying Expressions

Simplifying an algebraic expression is the process of transforming it into its simplest form, making it easier to work with. This is an essential skill in algebra as it helps solve problems efficiently.
To simplify the expression from the problem, we look at \( -\frac{5}{x} - -\frac{2}{x} \). Here, 'subtracting a negative' becomes an addition, simplifying to \( -\frac{5}{x} + \frac{2}{x} \).
To further simplify, combine the fractions to get \( -\frac{3}{x} \). Simplifying expressions helps in understanding the underlying algebraic relationships better and makes subsequent operations more straightforward.

Translating Phrases into Algebraic Expressions

Translating phrases into algebraic expressions involves converting words into mathematical symbols and operations. This skill is crucial for problem-solving in algebra, where many problems are initially presented in a verbal or textual format.
For instance, in the given exercise, "the quotient of -2 and a number" is translated to \( -\frac{2}{x} \), while "subtracted from" indicates the operation needed between two elements.
Learning to translate phrases accurately helps in constructing correct algebraic expressions, which is fundamental to solving equations and understanding algebraic concepts. By mastering this skill, students can approach algebra problems with confidence, knowing they can reliably interpret the language into workable math.