Lesson 4: Real World Problems Using Inequalities

Lesson 4: Real World Problems Using Inequalities Key Words in Real World Problems that Involve Inequalities Example 1 Keith must rent a truck for the day to clean up the house and yard. Home Store Plus charges $20 for use of the truck, plus $0.59 a mile. Keith has no more than $100 to spend on the truck rental. Write an inequality that represents Keith s situation. What is the maximum number of miles that Keith can drive the truck without going over budget?

Example 2 In college, the average of four major test grades determines your final grade for the class. Joe received the following scores on his first three tests: 90, 75, 82. He would like at least a B (average of 80) on his grade report and he has one more test to take. Write an inequality that represents this situation. What is the minimum score that Joe must earn on the final test in order to have a B average? Justify your answer.

4. Skate land charges a $75 flat fee for birthday rental and $5.50 a person for skates, pizza, and birthday cake. Joanna has a budget of $175 for the birthday party expenses. Write an inequality that represents Joanna s situation and that does not exceed her budget. What is the maximum number of people that Joanna can invite to her birthday party? 5. Jesse is saving money to buy a new stereo system for his car. He makes $15.50 a week babysitting his cousins. He also received $150 for his recent birthday. He needs to save at least $315 to buy the stereo. Write an inequality that represents Jesses situation. What is the minimum number of weeks that Jesse will have to babysit in order to buy the stereo? 6. Ashley s scores on the first three math tests were 80, 65, and 92. There will be four tests. She needs an average of at least 80 to get a B on her report card. Write an inequality to represent this situation. What is the minimum score Ashley must earn on her final test in order to get a B average? Justify your answer.

7. Jill is planning to build a flower bed in her front yard. She wants the length to be twice as long as the width. The perimeter of the flower bed must be less than 180m. Write an inequality to represent this situation. What is the longest possible length (whole number) for the flower bed? 8. You are planning a child s birthday party. Party Palace charges $120 for room rental plus $6 per child. The most that you have to spend is $175. Write an inequality that represents this situation. What is the maximum number of children can attend the party based on your budget? Justify your answer. 9. Simple interest can be calculated using the following formula: I = Prt where I = Interest P = principal r = rate t = time in years Write an inequality that would help you to find the minimum amount of money that you should put into a savings account that pays 5% interest in order to earn at least $50 in interest at the end of 2 years. What is the minimum amount of money that you should invest in the savings account in order to earn at least $50 interest in 2 years? Justify your answer. Your next assignment will be a quiz on solving and graphing inequalities in one variable.

Lesson 4 Real World Problems Using Inequalities 1. Jason is installing a flower bed in his backyard. He wants the length to be 4 times the width. The overall perimeter cannot be greater than 32 feet. Write an inequality that represents Jason s situation. What is the maximum width of the flower bed? w = width (We are not given any information about the width, so that will be my variable) 4w = length (He wants the length to be 4 times the width) 32 = Perimeter (perimeter cannot be greater than 32 feet) *cannot be greater than means less than or equal to* Perimeter formula is: 2w + 2L = P 2w + 2L = P 2w + 2(4w) 32 2w + 8w 32 10w 32 Substitute your values into the perimeter formula. Simplify: 2(4w) = 8w Simplify: 2w + 8w = 10w 10w/10 32/10 Divide both sides by 10 w 32/10 or 3 & 1/5 feet. The inequality that represents this situation is: 2w + 2(4w) 32 which is simplified to: 10w 32 The maximum width of the flower bed is 3 &1/5 feet. 2. An automotive repair shop charges $32 dollars an hour for labor plus $120 for parts. Judy s budget only allows for her to spend $200 on car repairs. What is the maximum number of hours that the car mechanic can work on Judy s car in order to not exceed her budget? The repair shop charges 32 an hour plus parts. Let x = the number of hours. 32x + 120 200 32x + 120 120 200 120 Since her budget is $200, the amount must be less than or equal to. Subtract 120 from both sides 32x 80 Simplify (200-120 = 80) 32x/32 80/32 Divide by 32 on both sides x 2.5 The maximum number of hours that a mechanic can work on Judy s car is 2.5 hours.

3. Chris wants to order DVD s off the internet. Each DVD costs $14.99 and shipping for the entire order is $9.99. Chris has no more than $100 to spend. Write an inequality that represents Chris s situation. How many DVD s can Chris order without exceeding his $100 limit? Justify your answer. The order is calculated as: 14.99 times the number of DVD s plus shipping. Let x = the number of DVD s. 14.99x + 9.99 100 Since Chris has no more than 100, we will use less than or equal to. 14.99x + 9.99 9.99 100 9.99 Subtract 9.99 from both sides 14.99x 90.01 Simplify (100-9.99 = 90.01) 14.99x /14.99 90.01/14.99 Divide by 14.99 on both sides x 6.00 Chris can order 6 Cd s or less without exceeding $100. Justify: 14.99x + 9.99 100 Original equation 14.99(6) + 9.99 100 Substitute the largest possible number, 6 for x. 99.93 100 This is a true statement; therefore my answer is correct. 4. Skate land charges a $75 flat fee for birthday rental and $5.50 a person for skates, pizza, and birthday cake. Joanna has a budget of $175 for the birthday party expenses. Write an inequality that represents Joanna s situation and that does not exceed her budget. What is the maximum number of people that Joanna can invite to her birthday party? The price for birthday rental at Skate land is: 5.50 times the number of people plus the 75 fee. Let x = the number of people invited. 5.50x + 75 175 Since Joanna s budget is 175, we have to use less than or equal to. 5.50x + 75 75 175 75 5.50x 100 5.50x/5.50 100/5.50 x 18.18 Justify: 5.5x + 75 175 Original equation 5.5 (18) + 75 175 Substitute 18 for x. 174 175 True statement The maximum number of people that Joanna can invite to her party is 18.

5. Jesse is saving money to buy a new stereo system for his car. He makes $15.50 a week babysitting his cousins. He also received $150 for his recent birthday. He needs to save at least $315 to buy the stereo. Write an inequality that represents Jesses situation. What is the minimum number of weeks that Jesse will have to babysit in order to buy the stereo? Jesse s savings is: 15.50 times the number of weeks he babysits plus his b-day money of 150 Let x = the number of weeks that Jesse babysits. 15.50x + 150 315 Since Jesse needs at least 315, we will use greater than or equal to. 15.50x + 150 150 315 150 Subtract 150 from both sides 15.50x 165 Simplify (315-150 = 165) 15.50x/15.50 165 / 15.50 Divide by 15.50 on both sides x 10.65 The minimum number of weeks that Jesse will have to babysit is 11 weeks. Since the number of weeks must be greater than or equal to 10.65, we need to round up in order to correctly answer the question. Justify: 15.50 x + 150 315 Original equation 15.50(11) + 150 315 Substitute 10 for x. 320.5 315

6. Ashley s scores on the first three math tests were 80, 65, and 92. There will be four tests. She needs an average of at least 80 to get a B on her report card. Write an inequality to represent this situation. What is the minimum score Ashley must earn on her final test in order to get a B average? Justify your answer. To find an average, we add up all the scores and divide by the number of scores that you added together. Since there will be four tests, Ashley s scores are: 80, 65, 92, and x. We will divide by 4. Let x = the fourth test score. (80+65+ 92+ x)/4 80 Since her average needs to be at least 80, we will use greater than or equal to (237 + x)/4 80 Simplify: 80+65+92 = 237 4( 237 + x) 80(4) Multiply by 4 on both sides to get rid of the fraction. 237 + x 320 Simplify 237 237 + x 320 237 x 83 Justify: (80+65+ 92+ x)/4 80 Original equation (80+65+ 92+ 83)/4 80 Substitute 83 for x 80 80 Ashley needs at least an 83 on her test to get a B. 7. Jill is planning to build a flower bed in her front yard. She wants the length to be twice as long as the width. The perimeter of the flower bed must be less than 180m. Write an inequality to represent this situation. What is the longest possible length (whole number) for the flower bed? w = width (We are not given any information about the width, so that will be my variable) 2w = length (She wants the length to be twice (2x) as long as the width) < 180 = Perimeter (perimeter is less than 180 m) Perimeter formula is: 2w + 2L = P 2w + 2L = P 2w + 2(2w) < 180 2w + 4w < 180 6w < 180 Substitute your values into the perimeter formula. Simplify: 2(2w) = 4w Simplify: 2w + 4w = 6w 6w/6 < 180/6 Divide both sides by 6 w < 30 m. The inequality that represents this situation is: 2w +2(2w) < 180 which is simplified to: 6w < 180 The longest possible length is 58m. The width must be less than 30 m, so 29 m is the largest width. The length is twice the width. Therefore, the longest possible length is 29(2) = 58 m.

8. You are planning a child s birthday party. Party Palace charges $120 for room rental plus $6 per child. The most that you have to spend is $175. Write an inequality that represents this situation. What is the maximum number of children can attend the party based on your budget? Justify your answer. The birthday party is charged as: $6 times the number of children plus $120 room rental fee. Let x = the number of children attending the birthday party. 6x + 120 175 The most you have to spend is 175, so we will use less than or equal to. 6x + 120 120 175 120 6x 55 6x/6 55/6 x 9.1 The maximum number of children that can attend the party is 9. Justify: 6x + 120 175 Original equation 6(9) + 120 175 Substitute 9 for x. 54 + 120 175 174 175 9. Simple interest can be calculated using the following formula: I = Prt where I = Interest P = principal r = rate t = time in years Write an inequality that would help you to find the minimum amount of money that you should put into a savings account that pays 5% interest in order to earn at least $50 in interest at the end of 2 years. What is the minimum amount of money that you should invest in the savings account in order to earn at least $50 interest in 2 years? Justify your answer. r = 5 % or.05 I = 50 T = 2 P =? Let p = principal prt = I P(.05)(2) 50 We use greater than or equal to because the Interest must be at least $50..1p 50 Simplify:.05(2) =.1.1p/.1 50/.1 P = 500 The minimum amount of money that you should invest in order to earn at least $50 in interest is $500. Justify: Prt I 500(.05)(2) 50 50 50

1. Yolanda is landscaping a garden in her back yard. The width of the garden is 3.5 feet shorter than the length. The perimeter of the garden must be less than 29 feet. (4 points) Write an inequality that could be used to find the length and width of the garden. Let x = the length of the garden. (We are not given information about the length, so we will designate this as our variable.) X 3.5 = the width of the garden (The width is 3.5 feet shorter than the length) 2l +2w = Perimeter Perimeter is less than 29 feet. 2x+ 2(x-3.5) < 29 Substitute for l and w. The perimeter is less than (<) 29. Find the length of the garden. 2x+ 2(x-3.5) < 29 Substitute for l and w. The perimeter is less than (<) 29. 2x+ 2x 7 < 29 Distribute the 2 throughout the parenthesis. 4x 7 < 29 4x-7+7 <29 +7 Simplify: 2x+2x= 4x Add 7 to both sides 4x < 36 Simplify: 29+7 = 36 4x/4 < 36/4 Divide by 4 on both sides x < 9 Simplify: 36/ 4= 9 The length must be less than 9 feet. Give two examples of a length and width that would be appropriate for Yolanda s garden. Justify your answers. Length: 8 feet Width: 4.5 feet The length must be less than 9 feet. Therefore, it could be 8 feet. If the length is 8 feet, then the width is 3.5 feet shorter, so 8 3.5 = 4.5 Justify: 2l + 2w < 29 2(8) +2(4.5) <29 16 + 9 <29 25 <29 Length: 8.5 feet Width: 5 feet The length must be less than 9 feet. Therefore, it could be 8.5 ft. If the length is 8.5 ft, then the width is 3.5 ft shorter, so 8.5-3.5 = 5 Justify: 2l + 2w < 29 2(8.5) + 2(5) < 29 17 + 10 <29 27 < 29

2. Kara has earned 97,480 frequent flier points on her credit card. She wants to cash in on roundtrip tickets for her family. Each roundtrip ticket costs 34,700 points and she must use 3800 points as a service fee for the transaction. (4 points) Write an inequality that could be used to determine how many tickets she can purchase with her frequent flier miles. RT points (# of rt tickets) + Service fee Total points accumulated Let x = the number of roundtrip tickets needed 34,700x + 3800 97,480 What is the maximum number of tickets that she can purchase? We can solve the inequality to find out the maximum number of tickets that can be purchased. 34700x + 3800 97480 34700 + 3800 3800 97480 3800 Subtract 3800 from both sides 34700x 93680 Simplify: 97480-3800 = 93680 34700x/34700 93680 / 34700 Divide by 34700 X 2.7 Simplify The maximum number of tickets that she can purchase is 2. Since x is less than or equal to 2.7, we know that she can only purchase 2 since you can t purchase part of a ticket. How many miles will she have left in her account after her purchase? In order to figure out how many miles she ll have left in her account after her purchase, we need to substitute our answer of 2 from above for x. 34700x + 3800 97480 34700(2) + 3800 97480 73200 97480 From our inequality above, we know that she uses 73200 points to purchase the 2 round trip tickets. Since she started with 97480, we can subtract to find the balance that she has left over. 97480-73200= 24280 She will have 24280 points left in her account after her purchase.