Problem 93 - Introductory and Intermediate Algebra for College Students 4th

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 simplified algebraic expression for the given English phrase is \(-3 \div x\).

Step 1: Translation to algebraic expressions

Translate each English phrase into an algebraic expression. 'The quotient of -2 and a number' means \(-2 \div x\). 'Subtracted from' implies usage of subtraction operation (-). 'The quotient of -5 and the number' can be written as \(-5 \div x\). Therefore, the entire phrase can be written as \(-5 \div x - (-2 \div x)\).

Step 2: Simplify the expression

To simplify the expression we will use the fact that subtracting a negative number is equivalent to adding a positive number (because \(-(-a) = a\)). Therefore, the expression simplifies to \(-5 \div x + 2 \div x\). We can combine these into a single fraction by adding the numerical values: \((-5 + 2) \div x = -3 \div x\).

Key Concepts

Translating English to Algebra

When faced with the task of turning English phrases into algebraic expressions, it is crucial to understand the common keywords which signal specific operations. For instance, the word ‘quotient’ indicates division, while ‘subtracted from’ suggests a subtraction operation. Let's look at an exercise that involves translating English to algebra:

Simplifying Algebraic Expressions

Once we translate our English phrase to algebraic expressions, it is essential to simplify them. Simplifying makes algebra more manageable and can also reveal the essence of the problem. To simplify, we look for like terms and combine them, we reduce fractions when possible, and we perform operations according to order of operations rules. Often, simplification can involve eliminating double negatives, as seen in the given exercise, which changes subtraction of a negative to addition.

Quotients in Algebra

Understanding quotients in algebra is key for working with division operations within algebraic expressions. A quotient represents the result of division. When translating English phrases involving quotients, we'll be using the division symbol or writing the numerator over the denominator. For example, 'The quotient of -2 and a number' is written in algebra as \( -2 \div x \). It’s also worth mentioning that quotients can be tricky to work with when simplifying, and they often involve finding a common denominator to combine terms effectively.

Algebraic Subtraction

Algebraic subtraction can introduce negative numbers into expressions, which can occasionally lead to confusion. When subtracting a negative number, as in our exercise, it is equivalent to adding the positive number. It's important to recognize these situations so you can simplify the expression correctly. The correct understanding of algebraic subtraction allows us to move through simplification smoothly and reach the solution more straightforwardly.