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**An Introduction to Problem Solving**

OBJECTIVES 1. Write algebraic expressions that can be simplified 2. Apply the steps for problem solving

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**Consecutive Integer Problems**

© 2002 by Shawna Haider

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**Choose something to represent the first integer---let’s choose “x”.**

x = first integer Then to represent a consecutive integer, that would mean the integer right after x or x+1. x +1 = second integer The sum of two consecutive integers is Find the integers. Means add together The hard part is done! x + x +1 = 137 Now solve for x to get the first integer. Just add 1 to get the second.

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**The sum of two consecutive integers is 65. What is the second number?**

Let x = the 1st number Let x + 1 = the 2nd number Sum means??? x = 32 x + 1 = 33 x + x + 1 = 65 = 65 2x + 1 = 65 -1 -1 2x = 64

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**The sum of two consecutive integers is 27. What are the two integers?**

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**What two consecutive integers have a sum of 39?**

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**The sum of two consecutive integers is 9. What are the two integers?**

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**Let’s consider a problem that asked for consecutive even integers**

Let’s consider a problem that asked for consecutive even integers. Your first integer will still be “x”. x = first integer Then to represent a consecutive even integer, you would need to add 2 instead of 1 and get x+2. x +2 = second integer The sum of two consecutive even integers is Find the integers. Now you are ready to solve. x + x +2 = 626

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**Let’s consider a problem that asked for consecutive odd integers**

Let’s consider a problem that asked for consecutive odd integers. Your first integer will still be “x”. x = first integer Now what would you do to x to get to the next odd integer? Most students initial reaction is “add 1” but try x = 3 (an odd integer) and see what happens when you add 1. Not an odd integer. So what would you add to 3 to get the next odd integer? x +2 = second integer So whether the problem says even integer or odd integer, the setup would look the same. If x happens to be odd then when you add 2 you will be at the next odd integer and if it happens to be even and you add 2 you will be at the next even integer.

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**The perimeter of a rectangle**

with sides of length x and 2x - 1 The perimeter of a rectangle is the sum of the lengths of the sides x In words: sides sides 2 sides 2 sides 2x - 1 Translate: 2(x) (2x-1) Then: x x-2 6x simplify

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**General Strategy for Problem Solving**

UNDERSTAND the problem. During this step, become comfortable with the problem. Some ways of doing this are: Read and reread the problem. Choose a variable to represent the unknown. Construct a drawing. Propose a solution and check. Pay careful attention to how you check your proposed solution. This will help when writing an equation to model the problem.

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**TRANSLATE the problem into an equation.**

SOLVE the equation. INTERPRET the results: Check the proposed solution in the stated problem and state your conclusion.

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**FINDING THE ORIGINAL PRICE OF A COMPUTER**

Suppose that a computer store just announced an 8% decrease in the price of a particular computer model. If this computer sells for $2162 after the decrease, find the original price of this computer. What are the steps to solving this problem?

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**UNDERSTAND. Read and reread the problem**

UNDERSTAND. Read and reread the problem. Recall that a percent decrease means a percent of the original price. Let’s guess that the original price of the computer is $ The amount of decrease is then 8% of $2500, or (0.08)($2500) = $200. This means that the new price of the computer is the original price minus the decrease, or $ $200 = $ Our guess is incorrect, but we now have an idea of how to model this problem

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**2. TRANSLATE: In words: Original price of computer minus**

8% of original price is new price Translate: x - 0.08 x = 2160

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**0.92x = 2162 Divide both sides of equation**

3. SOLVE the equation x – 0.08x = 2162 0.92x = Divide both sides of equation x = Solution

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4. INTERPRET. Check: If the original price of the computer was $2350, the new price is $2350 – (0.08)($2350) = $ $188 = $2162 State: The original price of the computer was $2350

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