PRACTICE EXAM UNIT #6: SYSTEMS OF LINEAR INEQUALITIES

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1 _ School: Date: /45 SCAN OR FAX TO: Ms. Stamm (Central Collegiate) FAX: (306) PH: (306) PRACTICE EXAM UNIT #6: SYSTEMS OF LINEAR INEQUALITIES Multiple Choice Identify the choice that best completes the statement or answers the question. 1. (1 point) For which inequality is ( 5, 1) a possible solution? a. y > 9 b. y 2x 10 c. y 9 + 2x d. y < x 2 2. (1 point) What is the boundary line for the linear inequality 4x + 2y < 18? a. y = 18 2x b. y = 36 4x c. y = 9 2x d. x = 18 2y 3. (1 point) Which test point is in the solution set for the linear inequality {(x, y) 5x 2y 10, x R, y R}? a. (5, 2) b. (2, 5) c. (1, 0) d. (0, 1) 4. (1 point) How would you graph the solution set for the linear inequality 2y 2x 10? a. Draw a dashed boundary line y = x + 5, then shade below the line. b. Draw a dashed boundary line y = x + 5, then shade above the line. c. Draw a solid boundary line y = x + 5, then shade below the line. d. Draw a solid boundary line y = x + 5, then shade above the line. 5. (1 point) Which system of linear inequalities has no solution? a. 5x 5y > 0 5x + 5y > 0 b. 5x + 5y > 5 x + y > 0 c. x + y 5 x y 5 d. 5x + 2y > 0 2x + 5y > 0 1

2 6. (1 point) Identify the point of intersection for the following system of linear inequalities. {2y 6x < 12, 4x + 4y 8, x I, y I} a. ( 3, 1) b. ( 1, 3) c. (3, 1) d. (1, 3) 7. (1 point) Describe the boundary lines for the following system of linear inequalities. {2y 6x < 12, 4x + 4y 8, x I, y I} a. Dashed line along y = 3x + 6; dashed line along y = 2 x b. Dashed line along y = 3x + 6; solid line along y = 2 x c. Solid line along y = 3x + 12; dashed line along y = 2 x d. Solid line along y = 3x + 12; solid line along y = 2 x 8. (1 point) Which test point is in the solution set for the following system of linear inequalities? {2y 6x < 12, 4x + 4y 0, x I, y I} a. (1, 2) b. (2, 1) c. ( 10, 0) d. ( 1, 1) 9. (1 point) What system of linear inequalities is shown here? a. y + 2x 3 y > 2x + 3 b. y 2x 2 y > 2x 2 c. y x 3 y > x 3 d. y 2x 3 y > 2x 3 2

3 10. (1 point) A vending machine sells juice and pop. The machine holds, at most, 200 cans of drinks. Sales from the vending machine show that at least 3 cans of juice are sold for each can of pop. Each can of juice sells for $1.50, and each can of pop sells for $1.00. Let x represent the number of cans of pop. Let y represent the number of cans of juice. How would you write the objective function for revenue, R? a. R = x y b. R = 1.25x + y c. R = 1.50(x + y) d. R = 1.50y x 11. (1 point) A vending machine sells juice and pop. The machine holds, at most, 200 cans of drinks. Sales from the vending machine show that at least 3 cans of juice are sold for each can of pop. Each can of juice sells for $1.50, and each can of pop sells for $1.00. Let x represent the number of cans of pop. Let y represent the number of cans of juice. Which of the following is a constraint of this optimization problem? a. 3x y b. x 3y c. x 3y d. 3x y 12. (1 point) Jan volunteers to fold origami frogs and swans for a display. She has 8 squares of green paper for the frogs and 12 squares of white paper for the swans. It takes her 4 min to fold an origami frog and 3 min to fold an origami swan. There must be two swans for every frog. Let f represent the number of frogs. Let s represent the number of swans. Which of the following is a constraint for this situation? a. f = 2s b. f > 2s c. 2f = s d. 2f < s 13. (1 point) Which location best describes where would you find the optimal solutions to an objective function? a. outside the feasible region b. at or near the points of intersection c. within the feasible region d. along a boundary line 3

4 14. (1 point) Where might you find the maximum solution to the objective function? Restrictions: x R y R Constraints: 2 x 4 2 y 4 Objective function: B = 2y + 3x a. (4, 2) b. (4, 4) c. ( 2, 4) d. ( 2, 2) 15. (1 point) Brent found spiders and grasshoppers in his barn. There were at most 12 spiders and at least 10 grasshoppers. There were no more than 36 spiders and grasshoppers, in total. Let s represent the number of spiders and let g represent the number of grasshoppers. Which inequality represents a restriction of s and g based on the given information? a. s + g > 36 b. s g 36 c. s g 22 d. s + g (1 point) Audrey notices the number of people and dogs in a dog park. There are more people than dogs. There are at least 12 dogs. There are no more than 40 people and dogs, in total. Let d represent the number of dogs and let p represent the number of people. Which inequality represents a restriction of d and p based on the given information? a. d p 40 b. d p 12 c. d < p d. 2d p Short Answer 17. (2 points) Why would you use a dashed boundary line when graphing the solution set of the linear inequality 1.6x 3y < 50? 4

5 18. (2 points) Is the point ( 2, 2) in the solution set for the linear inequality 4y 2x 0? 19. (2 points) Graph the system of linear inequalities: {(x, y) x + y 2, x > 3, x R, y R} 20. (2 points) Is the point (0, 5) in the solution set for the following system of linear inequalities? {y 2x 2, y > 3x 5, x R, y R} 21. (2 points) A student council is ordering signs for the winter dance. Signs can be made in letter size or poster size. No more than 30 of each size are wanted. No more than 50 signs are needed altogether. Letter-size signs cost $8.75 each, and poster-size signs cost $14.50 each. Let l represent the number of letter-size signs. Let p represent the number of poster-size signs. Write the objective function to determine the combination of the two sizes of signs that would result in the lowest cost to the council. 5

6 22. (2 points) A system of linear inequalities has vertices at (2, 4), ( 2, 5), and (0, 0). Which point represent the maximum value of the objective function Z = 4y x? 23. (2 points) A butcher shop makes hamburger patties and sausages. Hamburger patties sell for $2.50 and sausage sell for $2. The butcher noticed that they always sell at least four times as many hamburger patties as sausages. The butcher never sells more than 1000 hamburger patties. Let h represent the number of hamburger patties sold. Let s represent the number of sausages sold. Write a system of linear inequalities to describe the constraints. Then, write an objective function that represents the profit made from the sale of hamburger patties and sausages. 6

7 Problem 24. (3 points) Henri coaches a women s lacrosse team of 12 players. He plans to buy new practice jerseys and lacrosse sticks for the team. The supplier sells practice jerseys for $55 each and lacrosse sticks for $75 each. Henri can spend no more than $1700 in total. He wants to know how many jerseys and sticks he should buy. a) Write a linear inequality to represent the situation. b) Use your inequality to model the situation graphically. c) Determine a reasonable solution to meet the needs of the team, and provide your reasoning. 7

8 25. (3 points) Odette is setting up her social networking page: She wants to have no more than 460 friends on her new social networking page. She also wants to have at least two school friends for every karate friend. a) Define the variables and write a system of inequalities that models this situation. b) Describe the restrictions on the domain and range of the variables. 26. (3 points) A football stadium has seats. 65% of the seats are in the lower deck. 35% of the seats are in the upper deck. At least tickets are sold per game. A lower deck ticket costs $75, and an upper deck ticket costs $40. The management wants to maximize the revenue each game. a) Create a model to represent this situation. b) Suppose that cost of each ticket increased by $10. How would your model change? 8

9 27. (5 points) A refinery produces oil and gas. At least 1.5 L of gasoline are produced for each litre of heating oil. The refinery can produce up to 8.5 million litres of heating oil and 4 million litres of gasoline each day. Gasoline is projected to sell for $1.05 per litre. Heating oil is projected to sell for $1.90 per litre. The company needs to determine the daily combination of gas and heating oil that must be produced to maximize revenue. Create a model to determine this combination. What would the revenue be? Optimization Model Let g represent the number of millions of litres of gasoline. Let h represent the number of millions of litres of heating oil. Let R represent the total revenue from sales in millions of dollars. Restrictions: g R, h R Constraints: g 0 h 0 g 1.5h g 4 h 8.5 Objective function to maximize: R = 1.05g h 9

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11 ID: A PRACTICE EXAM UNIT #6: SYSTEMS OF LINEAR INEQUALITIES Answer Section MULTIPLE CHOICE 1. B 2. C 3. A 4. D 5. A 6. B 7. B 8. D 9. D 10. A 11. A 12. C 13. B 14. B 15. D 16. C SHORT ANSWER 17. To indicate that the points on the boundary line are not part of the solution set. 18. no no 21. C = 8.75l p 22. ( 2, 5) 1

12 ID: A 23. Constraints: h 0 s 0 h s h Objective function: P = 2.5h + 2s PROBLEM 24. a) Let x represent the number of jerseys. Let y represent the number of sticks. {(x, y) 55x + 75y 1700, x W, y W} b) Graph the line 55x + 75y = x-intercept: x = y-intercept: y = Since (0, 0) is in the solution set, the solution set is all points to the left of the line. c) e.g., Henri can buy 13 practice jerseys and 13 sticks for his team for $1690. It s reasonable to have an extra jersey and an extra stick. 25. a) Let x represent the number of school friends. Let y represent the number of karate friends. {(x, y) x + y 460, x W, y W} {(x, y) x 2y, x W, y W} b) The variables must be whole numbers (x W, y W) because she cannot count parts of people. 2

13 ID: A 26. a) Let x represent the number of lower deck tickets. Let y represent the number of upper deck tickets. Let R represent the total revenue. x W, y W x y x + y Objective function to maximize: R = 75x + 40y b) The new objective function to maximize would be: R = 85x + 50y 27. Use technology to graph the lines and find the intersection points of the solution area. Ê The intersection points are (0, 0), 4, 8 ˆ and (4, 0). Ë Á 3, Ê The maximum value of the objective function is at 4, 8 ˆ Ë Á 3. Ê R = 1.05(4) ˆ Ë Á 3 R = The maximum revenue is about $9.3 million. 3