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SREB UNIT 2 LESSON 5LESSON 5: INEQUALITIESDESCRIPTION: Students will explore the connection between equality and inequality. The behavior of inequalities in the negative number system is explored as well. ENGAGE/EXPLORE: TASK 13 EVAN AND MEGANEvan will be given a number between zero and 999. ?Evan multiplies the number by four and gives the result to Megan. ?Whenever Megan gets a number, she subtracts it from 2,000 and passes the result back to Evan. ?Evan multiplies that by four and passes the number back to Megan, etc. ?The winner is the last person who produces a number less than 1,000.?Break into pairs and record a couple of iterations of the game on a similar table: ?In the example above, Megan wins in round two since Evan produced a number greater than 1,000. ?In the example above, Megan wins in round one since Evan produced a number greater than 1,000. Thus we see in this case, large values cause Evan to lose whereas in the previous game, when Evan received a large number initially, he won. ?How can this situation be represented as an inequality? Work in your groups to set up and solve an inequality. ?Ask the pairs to find the largest number Evan could initially be given so that he would win in the first round. Solutions to Evan and Megan: 2000-4n > 1000-4n > -1000n < 250EXPLANATION: Observe students working on the Evan and Megan scenario. Pay particular attention to students who have n > 250 and n < 250. Have a group put one of each of the solutions on the board if they both exist in the class. Ask the class: Which is correct? Why? Make sure to think about the context of the game and the winner. Who should the winner be? Do your results show this? Why does the sign flip? (Students should observe that if the initial value is large (or greater than 250) then Evan will not win the first round. If the initial value is small (less than 250) Evan will win the first round. So it makes sense the sign is “flipped;” we want the values less than or equal to 250, n < 250. CLASS ACTIVITY/DISCUSSION: TASK 14 INEQUALITY BEHAVIORIn each case, describe what operations occurred to move from the direct, previous line. Using what you know about the structure of our number system, make a decision for the inequality symbol. What operations appear to be “flipping” the sign? What is true about the negative number system? Does adding and subtracting a negative number ALWAYS/SOMETIMES/NEVER produce an opposite number? Explain. Does multiplying or dividing by a negative number ALWAYS/SOMETIMES/NEVER produce an opposite number? Explain. ANSWERS:What operators appear to be “flipping” the sign? Multiplying and dividing with a negative value. What is true about the negative number system? The negative number system moves towards the left instead of towards the right. For example -1 > -2 where as 1 < 2 in positive numbers. This is due to the structure of the number line and the placement of numbers left to right on the real number line. Does adding and subtracting a negative number ALWAYS/SOMETIMES/NEVER produce an opposite number? Explain. Adding and subtracting with a negative number sometimes produces a negative number, but not always. The number will remain positive if the original number was positive and larger in magnitude than the negative value. Does multiplying or dividing by a negative number ALWAYS/SOMETIMES/NEVER produce an opposite number? Explain. Multiplying and dividing will always produce an opposite result. Multiplying (or dividing) a negative by a negative will produce a positive and likewise multiplying a positive by a negative will produce a negative number. PRACTICE: TASK 15 FISHING ADVENTURESFishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry 1,200 pounds (lbs) of people and gear for safety reasons. Assume the average weight of a person is 150 pounds. Each group will require 200 pounds of gear for the boat plus 10 pounds of gear for each person. a)Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set. ?b)Several groups of people wish to rent a boat. Group one has four people. Group two has five people. Group three has eight people. Which of the groups, if any, can safely rent a boat? What is the maximum number of people that may rent a boat? ?Possible Solutions: a. Let p be the number of people in a group that wishes to rent a boat. Then 150p represents the total weight of the people in the boat, in pounds. Also, 10p represents the weight of the gear that is needed for each person on the boat. So the total weight in that boat that is contributed solely by the people is 150p + 10p = 160p. Because each group requires 200 pounds of gear regardless of how many people there are, we add this to the above amount. We also know that the total weight cannot exceed 1200 pounds. So we arrive at the following inequality: 160p + 200 ≤ 1200. A graph illustrating the solutions is shown below. We observe that our solutions are values of p, listed below the number line and shown by the blue dots, so that the corresponding weights 160p + 200, listed above the line, are below the limit of 1200 lbs. b. We can find out which of the groups, if any, can safely rent a boat by substituting the number of people in each group for p in our inequality. We see that For Group 1: 160(4) + 200 = 840 ≤ 1200 For Group 2: 160(5) + 200 = 1000 ≤ 1200 For Group 3: 160(8) + 200 = 1480 ≤ 1200 We find that both Group 1 and Group 2 can safely rent a boat, but that Group 3 exceeds the weight limit, and so cannot rent a boat. To find the maximum number of people that may rent a boat, we solve our inequality for p: 160p + 200 ≤ 1200 160p ≤ 1000 p≤ 6.25 As we cannot have 0.25 person, we see that 6 is the largest number of people that may rent a boat at once. This also matches our graph; since only integer values of p make sense, 6 is the largest value of p whose corresponding weight value lies below the limit of 1200 lbs.TASK 16: SPORTS EQUIPMENT SETJonathan wants to save up enough money so that he can buy a new sports equipment set that includes a football, baseball, soccer ball, and basketball. This complete boxed set costs $50. Jonathan has $15 he saved from his birthday. In order to make more money, he plans to wash neighbors’ windows. He plans to charge $3 for each window he washes, and any extra money he makes beyond $50 he can use to buy the additional accessories that go with the sports box set. Write and solve an inequality that represents the number of windows Jonathan can wash in order to save at least the minimum amount he needs to buy the boxed set. Graph the solutions on the number line. What is a realistic number of windows for Jonathan to wash? How would that be reflected in the graph? Possible Solutions: We wish to find out how many windows Jonathan must wash, so let w be the number of windows. As he expects to get $3 per window, we multiply these two quantities. 3w This represents how much money Jonathan will make just from his window washing. Since he already has $15 saved, we now add 15 to this amount. 3w + 15 Because we know that Jonathan needs a minimum of $50, but could have more, we set this greater than or equal to 50. 3w + 15 ≥ 50 ?We can solve this expression by first subtracting 15 from both sides, and then dividing both sides by 3 to isolate w. CLOSING ACTIVITY: TASK 17: BASKETBALLChase and his brother like to play basketball. About a month ago, they decided to keep track of how many games they have each won. As of today, Chase has won 18 out of the 30 games against his brother. How many games would Chase have to win in a row in order to have a 75% wining record? ?How many games would Chase have to win in a row in order to have a 90% winning record? ?Is Chase able to reach a 100% winning record? Explain why or why not. ?Suppose that after reaching a winning record of 90% in part (b), Chase had a losing streak. How many games in a row would Chase have to lose in order to drop down to a winning record below 55%? ? So Chase will need to lose the next 77 games in order for his win percentage to drop below 55%. TASK 18: SOLVING INEQUALITIESSolve each of the following. Explain each step in your work, and check your answers. Jane plans to purchase three pairs of slacks all costing the same amount, and a blouse that is $4 cheaper than one of the pairs of slacks. She has $75 to spend but wants to have at least $3 left. What is the price range for the slacks? ?(-3x + 7) - 4(2x - 6) - 12 > 7 ?-3(5x - 3) < 4(x + 3) - 12 ? ................
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