Linear Equations and Inequalities in One Variable

An Introduction to Problem Solving

This section introduces strategies for translating real-world problems into algebraic equations and solving them using linear equations in one variable. Mastery of these skills is essential for tackling word problems and understanding mathematical modeling in College Algebra.

• Translation of English Phrases: Learn to convert verbal statements into algebraic expressions using addition, subtraction, multiplication, and division.

• Solving Word Problems: Apply systematic strategies to set up and solve equations based on problem conditions.

Algebraic Translations of English Phrases

Translating English phrases into algebraic expressions is a foundational skill in algebra. Each operation has characteristic language cues:

• Addition: Words like "sum," "increased by," "more than," and "plus" indicate addition. Example: "A number increased by 5" becomes .

• Subtraction: Phrases such as "difference," "decreased by," "less than," and "minus" signal subtraction. Example: "Four less than a number" is .

• Multiplication: "Product," "times," "twice," and "of" often mean multiplication. Example: "Six times a number" is .

• Division: "Quotient," "divided by," and "per" suggest division. Example: "A number divided by 3" is .

• More Than One Operation: Complex phrases may require combining operations. Example: "Four subtracted from six times a number" is .

Strategy for Solving Word Problems

Solving word problems involves a structured approach to ensure accuracy and clarity:

1. Read Carefully: Understand what is given and what is required. Restate the problem in your own words.

2. Assign Variables: Let (or another variable) represent an unknown quantity.

3. Write Expressions: Express other unknowns in terms of if needed.

4. Model with an Equation: Translate the problem conditions into an equation.

5. Solve and Check: Solve the equation, answer the question, and verify the solution in the context of the original problem.

Examples of Word Problem Solutions

Applying the above strategies, here are several examples:

Example 1: Solving a Word Problem

Problem: Four subtracted from six times a number is 68. Find the number.

• Let = the number.

• Equation:

• Solution steps:

• Answer: The number is 12.

Example 2: Consecutive Integers

Consecutive integers are numbers that follow each other in order, differing by 1.

• Problem: Two facing pages of a book have a sum of 145. What are the page numbers?

• Let = first page number, = second page number.

• Equation:

• Solution steps:

• Answer: The page numbers are 72 and 73.

Example 3: Taxi Ride

Problem: A taxi charges $2.00 to turn on the meter plus $0.25 for each eighth of a mile. If you have $10.00, how many eighths of a mile can you go? How many miles is that?

• Let = number of eighths of a mile.

• Equation:

• Solution steps:

  • Convert to miles: miles

• Answer: You can go 32 eighths of a mile, which is 4 miles.

Example 4: Price Reduction

Problem: After a 40% price reduction, an exercise machine sold for $564. What was the original price?

• Let = original price.

• Equation:

• Solution steps:

• Answer: The original price was $940.

Summary Table: Common Algebraic Translations

The following table summarizes common English phrases and their algebraic translations:

Additional info: The image included is the cover of an introductory algebra textbook, which is directly relevant as it visually represents the subject matter and context for College Algebra students.