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**An Introduction to Problem Solving**

§ 2.5 An Introduction to Problem Solving

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**Strategy for Problem Solving**

General Strategy for Problem Solving UNDERSTAND the problem Read and reread the problem Choose a variable to represent the unknown Construct a drawing, whenever possible Propose a solution and check TRANSLATE the problem into an equation SOLVE the equation INTERPRET the result Check the proposed solution in the problem State your conclusion

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**Finding an Unknown Number**

Example: The product of twice a number and three is the same as the sum of five times the number and 12. 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 sum of five times the number and 12” translates to 5x + 12. Continued

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**Finding an Unknown Number**

Example continued: 2.) TRANSLATE The product of the sum of + is the same as = twice a number 2x 5 times the number 5x and 3 3 and 12 12 Continued

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**Finding an Unknown Number**

Example continued: 3.) SOLVE 2x · 3 = 5x + 12 6x = 5x Simplify left side. 6x + (– 5x) = 5x + (– 5x) Add –5x to both sides. x = Simplify both sides. 4.) INTERPRET Check: Replace “number” in the original statement of the problem with 12. The product of twice 12 and 3 is 2(12)(3) = 72. The sum of five times 12 and 12 is 5(12) + 12 = 72. We get the same results for both portions. State: The number is 12.

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**Solving a Problem 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? 1.) UNDERSTAND Read and reread the problem. Let’s propose that we drive a total of 100 miles over the 2 days. Then we need to take twice the daily rate and add the fee for mileage to get 2(24.95) (100) = = This gives us an idea of how the cost is calculated, and also know that the number of miles will be greater than If we let x = the number of whole miles driven, then 0.29x = the cost for mileage driven Continued

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**Solving a Problem Example continued: 2.) TRANSLATE Daily costs**

2(24.95) mileage costs 0.29x 100 maximum budget plus + is equal to = Continued

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**Solving a Problem Example continued: 3.) SOLVE 2(24.95) + 0.29x = 100**

x = Simplify left side. Subtract from both sides. 49.90 – x = 100 – 49.90 0.29x = Simplify both sides. Divide both sides by 0.29. x Simplify both sides. Continued

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**Solving a Problem Example continued: 4.) INTERPRET**

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.

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EXAMPLE 1 Solve an equation with variables on both sides 7 – 8x = 4x – 17 7 – 8x + 8x = 4x – 17 + 8x 7 = 12x – 17 24 = 12x Write original equation. Add.

EXAMPLE 1 Solve an equation with variables on both sides 7 – 8x = 4x – 17 7 – 8x + 8x = 4x – 17 + 8x 7 = 12x – 17 24 = 12x Write original equation. Add.

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