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Money, Mixture, Motion, and Inequalities I. Homework II. Money Problems Example: You have 40 coins in nickels and dimes, How many dimes do you have if you have a total of $2.85? Let d = the number of dimes you have, then 40 - d = the number of nickels that you have. The total money that you have is 10d + 5(40 - d) = 285 10d + 200 - 5d = 285 5d + 200 = 285 5d = 85 d = 17
You are the manager of the new Tahoe Stadium. You sell your VIP seats for $200 each and your general admission seats for $75. Your stadium holds 100,000 people, and you need to earn at least $10,000,000. If you sell out, how many of your seats should you designate as VIP seats? Solution: Let x = number of VIP seats then 100,000 - x = number of general admission seats. The money from the VIP seats is 200x and the money from the general admission seats is 75(100,000 - x) Hence 200x + 75(100,000 - x) = 10,000,000 200x + 7,500,000 - 75x = 10,000,000 125x + 7,500,000 = 10,000,000 125x = 2,500,000 x = 20,000 We designate 20,000 seats as VIP seats. III. Mixture Problems Vodka contains 40% alcohol and wine contains 10% alcohol. You want to make a new drink that is 20% alcohol using vodka and wine. How much of each should you use to make 15 ounces of this drink? Solution: We let x = number of ounces of vodka. then 15 - x is the number of ounces of wine Note that the amount of alcohol is 15(.2) = 3 Hence we can write .4x + .1(15 - x) = 3 Multiplying by 10 we get: 4x + (15 - x) = 30 4x + 15 - x = 30 3x = 15 x = 5 Hence we pour 5 ounces of vodka and 10 ounces of wine. (It is not recommended to try this at home). IV. Motion Problems Example: Suppose that I am walking from school at 3 miles per hour and start at 12:00. At 12:30, you start riding your bike at 18 miles per hour to find me. At what time do you find me? Solution: Let t = the time after 12:00 We use the formula distance = rate times time Then my distance from school is 3t Your distance from school is 18(t - 1/2) We set the two equal to each other: 3t = 18(t - 1/2) distributing through, 3t = 18t - 9 subtracting 18t from both sides, -15t = -9 dividing both sides by -15, t = 9/15 = 3/5 = 36/60 or 36 minutes. Hence you find me at 12:36.
V. Linear Inequalities Definition: A linear inequality is one that can be reduced to ax + b < 0 or ax + b > 0 or ax + b < 0 or ax + b > 0 Step by step method for solving linear inequalities: 1) Simplify both sides (distribute and combine like terms) 2) Bring the x's to the left and the constants to the right 3) divide by the coefficient (changing the inequality if the sign of the inequality is negative) 4) Plot on a number line (remember holes and dots)
Example: 2(x - 1) < 3(2x - 4) 2x - 2 < 6x - 12 -4x < -12 x > 3 Plot on a number line with a hole at 3 and an arrow to the right of 3. The class will try 1) 3x - 5 > 6(x - 1) 2) 5(x + 2) < 2(x - 3)
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