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Copyright Copyright © 2011 © 2011 Pearson Pearson Education, Education, Inc. Publishing Inc. Publishing as Prentice as Prentice Hall Hall Chapter 2 Equations, Inequalities, and Problem Solving Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 2.2 An Introduction to Problem Solving Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Consecutive, even, and odd integers and their representations. +2 +1 Consecutive Integers: x x+1 x+2 +4 Consecutive Even Integers: +2 x x+2 Consecutive Odd Integers: +4 +2 x Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall x+2 x+4 x+4 Example 1 Write the following as an algebraic expression. Then simplify. The sum of three consecutive odd integers, if x is the first consecutive integer. Solution first plus next odd plus next odd In words: integer integer integer Translate: x Then x + (x + 2) + (x + 4) = x + x + 2 + x + 4 = 3x + 6 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall + (x + 2) + (x + 4) General Strategy for Problem Solving 1) UNDERSTAND the problem. During this step, become comfortable with the problem. Some way of doing this are: • Read and reread the problem. • Propose a solution and check. • Construct a drawing. • Choose a variable to represent the unknown. 2) TRANSLATE the problem into an equation. 3) SOLVE the equation. 4) INTERPRET the result. Check the proposed solution in the stated problem and state your conclusion. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 3 Find three numbers such that the second is 3 more than twice the first number, and the third number is four times the first number. The sum of the three numbers is 164. Solution UNDERSTAND the problem. Read and reread the problem. If we let x = the first number, then 2x + 3 = the second number 4x = the third number continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 2. TRANSLATE In words: first number added to second number added to third number is 164 Translate: x + (2x + 3) + 4x = 164 3. Solve the equation. x + (2x + 3) + 4x = 164 x + 2x + 4x + 3 = 164 Remove parentheses. 7x + 3 = 164 Combine like terms. 7x = 161 Subtract 3 from both sides. x = 23 Divide both sides by 7. 4. INTERPRET. Here, we check our work and state the solution; x = 23, the second number 2x + 3 = 2(23) + 3 = 49, and the third number 4x = 4(23) = 92. continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Check: Is the second number 3 more than twice the first number? Yes, since 3 more than twice 23 is 46 + 3, or 49. Also, their sum, 23 + 49 + 92 = 164, is the required sum. State: The three numbers are 23, 49, and 92. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 6 Kelsey Ohleger was helping her friend Benji Burnstine study for an algebra exam. Kelsey told Benji that her last three latest art history quiz scores are three consecutive even integers whose sum is 264. Help Benji find the scores. Solution UNDERSTAND the problem. Read and reread the problem. We are looking for consecutive integers, let x = the first integer. Then x + 2 = the second consecutive even integer x + 4 = the third consecutive even integer. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall continued 2. TRANSLATE first In words: integer + second even integer + third even integer = 264 Translate: x + (x + 2) + (x + 4) = 264 3. SOLVE. x + (x + 2) + (x + 4) = 264 3x + 6 = 264 Combine like terms. 3x = 258 Subtract 6 from both sides. x = 86 Divide both sides by 3. 4. INTERPRET. If x = 86, then x + 2 = 86 + 2 or 88, and x + 4 = 86 + 4 or 90. continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Check: The numbers 86, 88, and 90 are three consecutive even integers. Their sum is 264, the required sum. State: Kelsey’s art history quiz scores are 86, 88 and 90. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall