2 Chapter Chapter 2 Equations, Inequalities and Problem Solving

An Introduction to Problem Solving Section 2.4 An Introduction to Problem Solving

Solving Direct Translation Problems Objective 1 Solving Direct Translation Problems

Writing Phrases as Algebraic Expressions Addition ( + ) Subtraction ( - ) Multiplication ( ∙ ) Division ( ÷ ) Equal Sign Sum Difference of Product Quotient Equals Plus Minus Times Divide Gives Added to subtracted from Multiply Into Is/was/ should be More than Less than Twice Ratio Yields Increased by Decreased by Of Divided by Amounts to Total Less Represents Is the same as Objective A

Strategy for Problem Solving

Example Twice a number plus 3 is the same as the number minus 6. Objective A

Example 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. Continued

Example (cont) 2. Translate The product of · the difference of – is the same as = twice a number 2x 5 times the number 5x and ¾ and 3 3 Continued

Example (cont) 3. Solve 2x · 3 = 5x – ¾ 6x = 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.

Solving Problems Involving Relationships Among Unknown Quantities Objective 2 Solving Problems Involving Relationships Among Unknown Quantities

Example 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) + 0.29x = 100 Continued

Example (cont) 2(24.95) + 0.29x = 100 49.90 + 0.29x = 100 Continued

Example (cont) 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 49.90 + 0.29(173) = 100.07, which is over our budget. However, 49.90 + 0.29(172) = 99.78, which is within the budget. State: The maximum number of whole number miles is 172.

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. Draw a diagram. Let x = degree measure of smallest angle 2x = degree measure of second angle 3x = degree measure of third angle Continued

Example (cont) Recall that the sum of the measures of the angles of a triangle equals 180. measure of second angle measure of third angle measure of first angle 180 equals x 2x + 3x + = 180 Continued

Example (cont) x + 2x + 3x = 180 6x = 180 x = 30 Check: If x = 30, then 2x = 2(30) = 60 and 3x = 3(30) = 90 The sum of the angles is 30 + 60 + 90 = 180. State: The smallest angle is 30º, the second angle is 60º, and the third angle is 90º.

Solving Consecutive Integer Problems Objective 3 Solving Consecutive Integer Problems

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 + 2 + x + 4 = 252 Continued

Example (cont) The sum of three consecutive even integers is 252. Find the integers. x + x + 2 + x + 4 = 252 3x + 6 = 252 3x = 246 The integers are 82, 84 and 86.