Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 9 Equations, Inequalities and Problem Solving

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 9.4 An Introduction to Problem Solving

Martin-Gay, Developmental Mathematics, 2e 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Writing Phrases as Algebraic Expressions Addition ( + ) Subtraction ( - ) Multiplication ( ∙ ) Division ( ÷ ) Equal Sign sum differenceproductquotientequals plus minus timesdividegives added to subtracted from multiplyinto is/was/ should be more than less thantwiceratioyields increased by decreased byofdivided byamounts to total less represents is the same as

Martin-Gay, Developmental Mathematics, 2e 44 Strategy for Problem Solving General Strategy for Problem Solving 1.UNDERSTAND the problem. Read and reread the problem. Choose a variable to represent the unknown. Construct a drawing. Propose a solution and check. 2.TRANSLATE the problem into an equation. 3.SOLVE the equation. 4.INTERPRET the result: Check proposed solution in problem. State your conclusion. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Example Twice a number plus 3 is the same as the number minus 6.

Martin-Gay, Developmental Mathematics, 2e 66 The product of twice a number and three is the same as the difference of five times the number and ¾. Find the number. 1. Understand Read and reread the problem. If we let x = the unknown number, then “twice a number” translates to 2x, “the product of twice a number and three” translates to 2x · 3, “five times the number” translates to 5x, and “the difference of five times the number and ¾” translates to 5x – ¾. Finding an Unknown Number Example continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 77 The product of · twice a number 2x2x and 3 3 is the same as = 5 times the number 5x5x and ¾ ¾ the difference of – 2. Translate continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e Solve 2x · 3 = 5x – ¾ 6x = 5x – ¾ x = –¾ 6x + (–5x) = 5x + (–5x) – ¾ 4. Interpret Check: Replace “number” in the original statement of the problem with –¾. The product of twice –¾ and 3 is 2(–¾)(3) = –4.5. The difference of five times –¾ and ¾ is 5(–¾) –¾ = – 4.5. We get the same results for both portions. State: The number is –¾. continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 99 A car rental agency advertised renting a Buick Century for $24.95 per day and $0.29 per mile. If you rent this car for 2 days, how many whole miles can you drive on a $100 budget? x = the number of whole miles driven, then 0.29x = the cost for mileage driven 2(24.95) x = 100 Example continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 10 2(24.95) x = x = x = x  – x = 100 – continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 11 Check: Recall that the original statement of the problem asked for a “whole number” of miles. If we replace “number of miles” in the problem with 173, then (173) = , which is over our budget. However, (172) = 99.78, which is within the budget. State: The maximum number of whole number miles is 172. continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 12 Example The measure of the second angle of a triangle is twice the measure of the smallest angle. The measure of the third angle of the triangle is three times the measure of the smallest angle. Find the measures of the angles. Let x = degree measure of smallest angle 2x = degree measure of second angle 3x = degree measure of third angle Draw a diagram. continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 13 Recall that the sum of the measures of the angles of a triangle equals 180. x2x2x+3x3x+=180 measure of first angle measure of second angle measure of third angleequals180 continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 14 x + 2x + 3x = 180 x = 30 If x = 30, then 2x = 2(30) = 60 and 3x = 3(30) = 90 The sum of the angles is = 180. Check: State:The smallest angle is 30º, the second angle is 60º, and the third angle is 90º. 6x = 180 continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 15 Example The sum of three consecutive even integers is 252. Find the integers. x = the first even integer x + 2 = next even integer x + 4 = next even integer Translate: x + x x + 4 = 252 continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 16 The sum of three consecutive even integers is 252. Find the integers. x + x x + 4 = 252 3x + 6 = 252 3x = 246 The integers are 82, 84 and 86. continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.