Problem 70 - 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 \(-15\) and a number, increased by the quotient of 4 and the number

Step-by-Step Solution

Answer

The algebraic expression for the given English phrase is \(\frac{-11}{x}\).

Step 1: Translate English phrases into algebraic expressions

The first phrase is 'The quotient of \(-15\) and a number'. This can be translated into algebra by dividing \(-15\) by \(x\), which is written as \(-\frac{15}{x}\). The second phrase is 'increased by the quotient of 4 and the number'. This can be expressed as adding the quotient of 4 and \(x\) to the first phrase, which is written as \(+\frac{4}{x}\). So, the combined phrase becomes \(-\frac{15}{x} + \frac{4}{x}\).

Step 2: Derive common denominator and simplify

Now, as the denominators of both fractions are the same, which is \(x\), you can add the numerators to simplify the expression. The simplified expression is then \(\frac{-15 + 4}{x}\).

Step 3: Further simplification

Combine the numbers in the numerator to simplify the expression further. The expression \(\frac{-15 + 4}{x}\) simplifies to \(\frac{-11}{x}\). So, the English phrase 'the quotient of \(-15\) and a number, increased by the quotient of 4 and the number' can be translated to the algebraic expression \(\frac{-11}{x}\).

Key Concepts

Quotient

A quotient in algebra is the result of dividing one number by another. When you see the word "quotient," it's a signal in algebra to form a division equation. Let's break this down with the example from the exercise:

• The expression \(-\frac{15}{x}\) represents the quotient of \(-15\) and some number, where \(x\) is the variable replacing the unknown number.
• Similarly, \(\frac{4}{x}\) is the quotient of \(4\) and the number \(x\).

When writing these in algebraic terms, it's important to ensure that the division is correctly represented. Always place the dividend, the number being divided (such as \(-15\) or \(4\)), on top, in the numerator position. The divisor, the number by which you are dividing (\(x\) in both cases here), goes in the denominator position.

Simplification

Simplification in algebra involves reducing expressions to their simplest form. This makes them easier to work with and understand. In the given exercise, after translating the English phrases into algebraic expressions, it leads to \[-\frac{15}{x} + \frac{4}{x} \]To simplify, notice that both terms have a common denominator \(x\). This allows us to combine the numerators:

• The expression becomes \(\frac{-15 + 4}{x}\).
• After performing the arithmetic operation \(-15 + 4\), which is \(-11\), it further simplifies to \(\frac{-11}{x}\).

These steps illustrate simplification involves combining like terms and performing basic arithmetic operations, like addition or subtraction, on the numerators while keeping the common denominator intact.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. It allows the fractions to be easily added or subtracted. In this exercise, both fractions involved already have the same denominator, \(x\). Therefore, there is no need to find an additional common denominator. Here's how it works:

• With both fractions having \(x\) as the denominator, we can directly apply arithmetic to their numerators.
• Combine numerators \(-15\) and \(4\) into a single expression like so: \(-15 + 4\).

This leads to keeping \(x\) as the denominator and simplifying the numerators as we have done before. When working with expressions, always ensure denominators are aligned before performing any addition or subtraction.