Problem 94 - 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 \(-7\) and a number, subtracted from the quotient of \(-12\) and the number.

Step-by-Step Solution

Answer

The equivalent algebraic expression for the given English phrase is \(-\frac{5}{x}\).

Step 1: Translate English phrases to algebraic expression

First, identify the word 'quotient' which means division. Translate the phrase 'the quotient of -7 and a number' into an algebraic expression, \(-\frac{7}{x}\). Similarly, translate 'the quotient of -12 and the number' to \(-\frac{12}{x}\). The phrase 'subtracted from' means that these expressions will be subtracting. The full algebraic expression becomes \(-\frac{12}{x} - (-\frac{7}{x})\).

Step 2: Simplify the algebraic expression

To simplify the expression \(-\frac{12}{x} - (-\frac{7}{x})\), change the double negative into a plus sign to get \(-\frac{12}{x} + \frac{7}{x}\). Then, combine like terms to produce the simplified expression \(-\frac{5}{x}\).

Key Concepts

Quotient in Algebra

In algebra, the term "quotient" refers to the result of dividing one number by another. Consider it as a way to express division as a standalone mathematical operation. For example, when you read "the quotient of -7 and a number," it translates to the algebraic expression \[-\frac{7}{x}\]where \(x\) represents the unknown number. Similarly, the phrase "the quotient of -12 and the number" becomes\[-\frac{12}{x}\]Both expressions share the common element of division, highlighting the central concept of quotients in algebra: they quantify how one value is distributed over another. This is crucial for translating real-world problems into mathematical equations. Understanding the notion of a quotient helps you differentiate division from other operations like multiplication or addition when working with complex algebraic problems.

Simplifying Expressions

The process of simplifying expressions in algebra is akin to tidying up an equation to make it more understandable or easier to work with. It's all about breaking down complicated expressions into simpler, clearer forms. Here's how you can simplify an expression, step by step.

• Start by evaluating any operations, like resolving a double negative or combining terms.
• In the expression \(-\frac{12}{x} - (-\frac{7}{x})\), first recognize the double negative, which simplifies to a positive.
• This step yields \(-\frac{12}{x} + \frac{7}{x}\), making it easier to see that you can now combine terms.
• Combine the fractions by subtracting \(\frac{7}{x}\) from \(-\frac{12}{x}\), resulting in \(-\frac{5}{x}\). This expression is now in its simplest form.

By following these steps, we can transform even complex expressions into something more manageable, making it an essential skill for solving algebraic problems efficiently.

Translating Phrases to Expressions

Translating phrases into algebraic expressions is a key skill in mathematics. It involves interpreting everyday language and expressing it using mathematical symbols and equations. Here's a strategy for translating phrases to expressions, especially when dealing with words that indicate mathematical operations.

• Identify keywords: Look for words like "divided by," "multiplied by," "sum," "difference," and notably, "quotient," which is vital for division.
• Align each part of the phrase with mathematical operations. "The quotient of -7 and a number" becomes \(-\frac{7}{x}\).
• Note any terms like "subtracted from," which indicate subtraction, as in "subtracted from the quotient of -12 and the number" leading to \(-\frac{12}{x} - \left(-\frac{7}{x}\right)\).
• Always check the sentence structure to assure the order of operations and relationships between numbers are accurately reflected in the expression.

By mastering this skill, students can accurately turn word problems into algebraic expressions, paving the way for successful simplification and problem-solving. It's not just about substituting numbers and operations, but understanding the math described by words.