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Chapter 2 Equations and Inequalities in One Variable

Chapter 2 Equations and Inequalities in One Variable. Section 5 Introduction to Problem Solving: Direct Translation Problems. Section 2.5 Objectives. 1 Translate English Phrases to Algebraic Expressions 2 Translate English Sentences to Equations

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Chapter 2 Equations and Inequalities in One Variable

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  1. Chapter 2 Equations and Inequalities in One Variable Section 5 Introduction to Problem Solving: Direct Translation Problems

  2. Section 2.5 Objectives 1 Translate English Phrases to Algebraic Expressions 2 Translate English Sentences to Equations 3 Build Models for Solving Direct Translation Problems

  3. Translating English Expressions

  4. Translating Sentences into Expressions Example: Translate each of the following into a mathematical statement. a.) Twelve more than a number is 25. x + 12 = 25 b.) One-third of the sum of a number and four yields 6.

  5. Problem Solving Problem solving is the ability to use information, tools, and our own skills to achieve a goal. The process of taking a verbal description of the problem and developing it into an equation that can be used to solve the problem is mathematical modeling. The equation that is developed is the mathematical model.

  6. Categories of Problems • Five Categories of Problems • Direct Translation – problems that must be translated from English into mathematics using key words in the verbal description • Mixtures – problems where two or more quantities are combined in some fashion • Geometry – problems where the unknown quantities are related through geometrical formulas • Uniform Motion – problems where an object travels at a constant speed • Work problems – problems where two or more entities join forces to complete a job

  7. Steps for Solving Problems Solving Problems with Mathematical Models Step 1: Identify What You Are Looking For Read the problem carefully. Identify the type of problem and the information we wish to learn. Typically the last sentence in the problem indicates what it is we wish to solve for. Step 2: Give Names to the Unknowns Assign variables to the unknown quantities. Choose a variable that is representative of the unknown quantity it represents. For example, use t for time. Step 3: Translate into the Language of Mathematics Determine if each sentence can be translated into a mathematical statement. If necessary, combine the statements into an equation that can be solved. Continued.

  8. Steps for Solving Problems Solving Problems with Mathematical Models Step 4: Solve the Equation(s) Found in Step 3 Solve the equation for the variable and then answer the question posed by the original problem. Step 5: Check the Reasonableness of Your Answer Check your answer to be sure that it makes sense. If it does not, go back and try again. Step 6: Answer the Question Write your answer in a complete sentence.

  9. Direct Translation Problem Example: In a baseball game, the Yankees scored 4 more runs than the White Sox. A total of 12 runs were scored. How many runs were scored by each team? Step 1: Identify.This is a direct translation problem. We are looking for the number of runs scored by each team. Step 2: Name.Let x represent the number of runs scored by the White Sox. The number of runs scored by the Yankees is equal to x + 4. Continued.

  10. Direct Translation Problem Example continued: Step 3: Translate.Since we know that the total number of runs is 12, we have White Sox runs Yankees runs x + x + 4 = 12 Step 4: Solve. x + x + 4 = 12 Combine like terms. 2x + 4 = 12 2x = 8 Subtract 4 from both sides. Divide both sides by 2. x = 4 Continued.

  11. Direct Translation Problem Example continued: Step 5: Check.Since x represents the number of runs scored by the White Sox, the White Sox scored 4 runs. The Yankees scored x + 4 = 4 + 4 = 8 runs. 4 + 8 = 12  Step 6: Answer the Question. The Yankees scored 8 runs and the White Sox scored 4 runs.

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