Chapter 11Inequalities and Linear Programming 1
Chapter 11Inequalities and Linear Programming
Warm-up Exercise1.Draw the graphs of the following equations.
(a)x2(b)y3
2.Draw the graphs of the following equations.
(a)yx(b)y2x1
3.Draw the graphs of the following equations.
(a)5x2y10(b)8x5y200
4.When x3 and y5, find the value of each of the following expressions.
(a)2x3y1(b)93x5y
5.Iff(x)5x3, find the values of the following.
(a)f(10)(b)f(4)
6.If, find the values of the following.
(a)f(2)(b)f(3)
7.What is the common feature of each of the following groups of straight lines?
(a)x3y0, x3y2, x3y6(b)x2y0, x2y10, x2y40
Build-up Exercise[ This part provides three extra sets of questions for each exercise in the textbook, namely Elementary Set, Intermediate Set and Advanced Set. You may choose to complete any ONE set according to your need. ]
Exercise 11A
Elementary Set
Level 1
Solve the following inequalities and represent the solutions graphically. (1 – 12)
1.2.
3.4.
5.6.
7.8.
9.10.
11.12.
Level 2
13.Find the least values of two consecutive odd numbers such that the smaller number is greater thanof the larger number.
14.Donald is 7cm taller than Joyce and Joyce is 15cm shorter than Kelvin. If the sum of their heights is not greater than 484cm, find the maximum height of Joyce.
Intermediate Set
Level 1
Solve the following inequalities and represent the solutions graphically. (15 – 24)
15.16.
17.18.
19.20.
21.22.
23.24.
Level 2
25.Find the largest values of two consecutive even numbers such that 2 times the sum of the larger number and 2 is not less than 3 times the smaller number.
26.(a)Write down three consecutive multiples of 6 if the smallest number is x.
(b)Hence find the greatest values of three consecutive multiples of 6 whose sum is less than 1000.
27.(a)Write down three consecutive multiples of 11 if the largest number is x.
(b)Hence find the least values of three consecutive multiples of 11 whose sum is greater than 1023.
28.A wire with a length of at most 98cm is bent to form a rectangle. If the width of the rectangle is 5cm longer than the length, find the maximum area of the rectangle.
Advanced Set
Level 1
Solve the following inequalities and represent the solutions graphically. (2936)
29.5(3x2)1530.
31.32.
33.34.
35.36.
Level 2
37.(a)Write down four consecutive multiples of 6 if the largest number is x.
(b)Hence find the least values of four consecutive multiples of 6 whose sum is not less than 180.
38.Mary is 15 years older than John. Four years later, Mary’s age will not be greater than twice of John’s. Find the minimum possible age of Mary at present.
39.One side of a rectangle is 32cm. If the perimeter of the rectangle is at most 100cm, find the maximum area of the rectangle.
40.Mr. Ho and Mr. Lee drive from city A to city B. Mr. Lee leaves city A 30 minutes earlier than Mr.Ho does. If Mr. Lee and Mr. Ho drive at 65km/h and 80km/h respectively, how long does Mr.Ho take to drive at least 65km ahead of Mr. Lee?
41.There are altogether 6 tests in a mathematics course. In order to obtain grade A at the end of the course, the average score of a student must not be less than 85. John got 79, 84, 90, 76 and 93 for the first 5 tests. What is the minimum score that John should get in the last test if he wants to obtain grade A at the end of the course?
42.A shop sells batteries in two packages A and B. Package A contains 8 batteries and package B contains 12 batteries. If Cathy needs at least 144 batteries and she will buy packages A and B in the ratio of 3:2, find the minimum number of packs of packages A and B that she will buy.
Exercise 11B
Elementary Set
Level 1
Solve the following simultaneous inequalities and represent the solutions graphically. (114)
1.1x and x32.
3.4.
5.6.
7.8.
9.10.
11.12.
13.14.
Level 2
Solve the following simultaneous inequalities and represent the solutions graphically. (1520)
15.16.
17.18.
19.12x3520.334x7
Intermediate Set
Level 1
Solve the following simultaneous inequalities and represent the solutions graphically. (2130)
21.22.
23.24.
25.26.
27.28.
29.30.
Level 2
Solve the following simultaneous inequalities and represent the solutions graphically. (3140)
31.32.
33.34.
35.36.
37.38.
39.35x717 40.
Advanced Set
Level 1
Solve the following simultaneous inequalities and represent the solutions graphically. (4148)
41.42.
43.44.
45.46.
47.48.
Level 2
Solve the following simultaneous inequalities and represent the solutions graphically. (4958)
49.50.
51.52.
53.54.
55.56.1210x635
57.58.
59.(a)Solve the simultaneous inequalities 1264x3.
(b)Find the integral solutions of the simultaneous inequalities in (a).
60.The sum of the lengths of any two sides of a triangle must be longer than the remaining side of the triangle. If the lengths of the three sides of a triangle are (3x4)cm, (302x)cm and (x4)cm,
(a)write down three inequalities satisfying the above condition.
(b)find the range of values of x.
Exercise 11C
Elementary Set
Level 1
Solve the following inequalities graphically. (16)
1.x3y62.xy80
3.4.2x5y0
5.3x4y126.7x2y14
Solve the following systems of inequalities graphically. (710)
7.8.
9.10.
Level 2
Solve the following systems of inequalities graphically. (1112)
11.12.
13.Write down the system of inequalities whose solutions are represented by the shaded region in the figure.
Intermediate Set
Level 1
Solve the following inequalities graphically. (1417)
14.4xy1215.2x4y100
16.17.4yx6
Solve the following systems of inequalities graphically. (1820)
18.19.
20.
Level 2
Solve the following systems of inequalities graphically. (2124)
21.22.
23.24.
25.Find all the integral solutions of the following system of inequalities graphically.
26.Referring to each of the following figures, write down the system of inequalities whose solutions are represented by the shaded region/dots.
(a)(b)
Advanced Set
Level 1
Solve the following inequalities graphically. (2728)
27.2x3y628.
Solve the following systems of inequalities graphically. (2930)
29.30.
Level 2
Solve the following systems of inequalities graphically. (3135)
31.32.
33.34.
35.
Find all the integral solutions of the following systems of inequalities graphically. (3637)
36.37.
38.Referring to each of the following figures, write down the system of inequalities whose solutions are represented by the shaded region/dots.
(a)(b)
(c)
39.(a)Referring to the figure, find the equations of the straight lines.
(b)Write down the system of inequalities whose solutions are represented by the shaded region.
Exercise 11D
Elementary Set
Level 1
1.x square tables and y rectangular tables are to be displayed in a function room without overlapping.A square table occupies a floor area of 2m2 and a rectangular table occupies a floor area of 5m2. Express the following conditions in terms of x and y.
(a)The function room can display at most 40 tables.
(b)The floor area of the function room is not greater than 150m2.
2.A restaurant makes xL of tomato soup and yL of chicken soup each day. The cost of making 1L of tomato soup is $9 and that of 1L of chicken soup is $15. Express the following conditions in terms of x and y.
(a)The cost of making the two kinds of soup is kept at most $600 each day.
(b)The total amount of soup made each day cannot be less than 50L.
3.A shop sells two packages of Christmas cards, A and B. The number of cards in each pack of A and B is as follows.
TraditionalChristmas cards / Pop-up
Christmas cards
Each pack of
package A / 3 / 2
Each pack of
package B / 4 / 1
If Jason needs at least 24 traditional Christmas cards and 8 pop-up Christmas cards, and he buys xpacks of package A and y packs of package B, write down all the constraints about x and y.
4.A worker produces wooden cupboards and wooden beds. The resources required to produce wooden cupboards and wooden beds are shown in the table.
Wood (unit) / Working hoursWooden cupboard / 16 / 21
Wooden bed / 12 / 36
If there are 120units of wood and 252 working hours available to produce x wooden cupboards and y wooden beds, write down all the constraints about x and y.
5.A candy shop produces two types of assorted candies, A and B. The distribution of candies in each box of A and B is as follows:
Lemon candies (pack) / Grape candies (pack) / Orange candies (pack)Each box of assorted candies A / 4 / 3 / 8
Each box of assorted candies B / 6 / 4 / 4
The candy shop has 140 packs of lemon candies, 100 packs of grape candies and 120 packs of orange candies to produce x boxes of A and y boxes of B. Write down all the constraints about x andy.
6.A pet shop wants to buy x cats and y dogs. A cat costs $250 and a dog costs $350. The shopkeeper has a budget of $6000 and she can only take care of at most 20 cats and dogs at one time. Write down all the constraints about x and y.
Level 2
7.A supplier orders two types of computers, desktop and notebook computers, subject to the following conditions:
(1)The number of desktop computers to be ordered each month should be at least 20 and at most 50.
(2)The number of notebook computers to be ordered each month should be at least 15 and at most 45.
(3)Due to the limited storage space, the total number of computers to be ordered cannot exceed 80per month.
Let x and y be the number of desktop computers and number of notebook computers to be ordered every month respectively.
(a)Write down all the constraints about x and y.
(b)Find the feasible solutions graphically.
8.A food manufacturer produces an animal feed from two grains A and B. Each kg of grain A contains 0.5unit of carbohydrates, 0.2unit of vitamins and 0.1unit of protein. Each kg of grain B contains 0.3unit of carbohydrates, 0.3unit of vitamins and 0.2unit of protein. In order to fulfill a nutritional standard, the mixture of these two grains must contain at least 2units of carbohydrates, 1.8units of vitamins and 1unit of protein. Let xkg and ykg be the weights of grains A and B required respectively.
(a)Write down all the constraints about x and y.
(b)Find the feasible solutions graphically.
Intermediate Set
Level 1
9.A car park provides parking space for private cars and trucks only. A private car requires a parking area of 12m2 while a truck requires a parking area of 21m2. Let x and y be the number of private cars and trucks parked in the car park respectively. Express the following conditions in terms of x and y.
(a)At most 100vehicles can be parked in the car park at the same time.
(b)The total parking area for private cars and trucks are 1500m2.
10.The amounts of vitamins B1, B2 and K in vitamin pills M and N are shown in the table.
Vitamin / B1 / B2 / KAmount in a vitamin pill M (unit) / 6 / 2 / 1
Amount in a vitamin pill N (unit) / 3 / 4 / 1
The minimum intake of vitamins B1, B2 and K for a person per week are 15 units, 12 units and 4units respectively. Let x and y be the number of vitamin pills M and N consumed per week by a person respectively. Write down all the constraints about x and y.
11.A factory produces two kinds of products, A and B. Each product Acosts $160 and requires 16man-hours in production. Each product B costs $270 and requires 8 man-hours in production. The budget is $12000 and there are 720man-hours available. Let x and y be the number of product A and number of product B to be produced. Write down all the constraints about x and y.
12.A rectangular lawn of xm long and ym wide is to be planted with flowers. The relevant data are as follows.
(1)The perimeter of the lawn should be between 20m and 46m.
(2)The length of the lawn should not be shorter than the width but should be at most 5 times the width.
Write down all the constraints about x and y.
Level 2
13.A dietician prescribes a diet of x kg of food X and y kg of food Y so that the resulting nutritional content is composed of at least 60 units of protein, 24 units of fats and 7.8 units of calories. The amounts of protein, fats and calories in each kg of food X and food Y are shown below:
Protein (unit) / Fats (unit) / Calories (unit)Each kg of food X / 50 / 12 / 4.8
Each kg of food Y / 40 / 24 / 6
(a)Write down all the constraints about x and y.
(b)Find the feasible solutions graphically.
14.A factory produces monitors in two sizes. A 15-inch monitor requires 4hours in final production and 2hours in testing. A 17-inch monitor requires 4hours in final production and 3hours in testing. It is known that 300working hours are available for final production and 180working hours are available for testing every day. Let x and y be the number of 15-inch monitors and number of 17-inch monitors to be produced every day respectively.
(a)Write down all the constraints about x and y.
(b)Find the feasible solutions graphically.
15.A merchant sells two different packages of magic clips. Package A contains 10small,6medium and 2large clips. Package B contains 4 small, 8 medium and 8large clips. Package A costs $20 each and package B costs $30 each. A customer wants to buy at least 40small, 30medium and 12large clips with at most $200. Let x and y be the number of packs of package A and package B respectively bought by the customer.
(a)Write down all the constraints about x and y.
(b)Find the feasible solutions graphically.
16.A ship can carry at most 1200kg of luggage and 140 passengers in a journey. Two classes of seats, first class and second class are provided. The number of first-class seats is not more than one-quarter of the number of second-class seats. The luggage of a first-class passenger and that of a second-class passenger cannot be heavier than 12kg and 8kg respectively. Let x and y be the number of first-class seats and number of second-class seats provided respectively.
(a)Write down all the constraints about x and y.
(b)Find the feasible solutions graphically.
Advanced Set
Level 1
17.A designer is going to renovate the rooms of a hostel which can provide accommodation for at least 130people. Each room requires $36000 for renovation and the budget for renovating all the rooms is $2800000 where the number of single rooms is not more than that of double rooms. If the hostel has x single rooms and y double rooms, write down all the constraints about x and y.
18.A baker makes two types of mooncakes, A and B. Each box of mooncake A needs 700g of flour and 200g of lotus seed paste. Each box of mooncake B needs 300g of flour and 500g of lotus seed paste. If 30kg of flour and 10kg of lotus seed paste are available for making the two types of mooncakes, x boxes of mooncake A and y boxes of mooncake B can be made. Write down all the constraints about x and y.
Level 2
19.The maximum load of luggage and maximum floor area for passengers on a train are 3000kg and 250m2 respectively. The train provides first-class seats and second-class seats which take up 2m2 and 1.4m2 of floor area per seat respectively. The number of first-class seats does not exceed one-sixth of the number of second-class seats. The luggage of a first-class passenger and a second-class passenger must not be heavier than 60kg and 15kg respectively. Let x and y be the number of first-class seats and the number of second-class seats respectively.
(a)Write down all the constraints about x and y.
(b)Find the feasible solutions graphically.
20.A factory manufactures cabinets and bookcases. A cabinet requires 2 hours for painting and 2hours for installation. A bookcase requires 1 hour for painting and 3 hours for installation. The factory has 10 painting machines and 10 installation machines. Each painting machine and installation machine is used for 9hours and for 12hours per day respectively. Let x and y be the number of cabinets and the number of bookcases to be produced every day respectively.
(a)Write down all the constraints about x and y.
(b)Find the feasible solutions graphically.
21.A hospital has to provide its patients with a diet containing at least 32units of carbohydrates, at least 15units of protein and at most 30 units of fats. Two types of food, A and B, are provided for the diet and their contents are as follows:
Carbohydrates (unit) / Protein (unit) / Fats (unit)Each g of food A / 0.96 / 0.35 / 0.2
Each g of food B / 0.4 / 0.25 / 0.25
If xg of food A and yg of food B are provided for the diet,
(a)write down all the constraints about x and y.
(b)find the feasible solutions graphically.
22.A factory manufactures furnitures A and B. Each piece of furniture A requires 20 man-hours and a material cost of $120. Each piece of furniture B requires 30 man-hours and a material cost of $90. There are 1800 man-hours available and a budget of $9000 for material. Let x and y be the number of furniture A and number of furniture B to be manufactured.
(a)Write down all the constraints about x and y.
(b)Find the feasible solutions graphically.
23.A manufacturer produces two types of juices, X and Y. Each litre of juice X requires 24g of material A, 18g of material B and 60g of material C. Each litre of juice Y requires 48g of material A, 10g of material B and 60g of material C. The manufacturer has 960g of material A, 360g of material B and 1440g of material C. If x litres of juice X and ylitres of juice Y are to be produced,
(a)write down all the constraints about x and y.
(b)find the feasible solutions graphically.
24.A factory produces boxes for digital video discs (DVD) and video compact discs (VCD). Each DVD box costs $5 in materials, $14 for packaging and $10 for labour. Each VCD box costs $3 in materials, $4 for packaging and $8 for labour. The expenditure on materials is controlled to be not more than $60 per hour and the expenditure on both packaging and labour are controlled to be not more than $140 per hour. If the factory produces x DVD boxes and y VCD boxes per hour,
(a)write down all the constraints about x and y.
(b)find the feasible solutions graphically.
Exercise 11E
Elementary Set
Level 1
In each of the following figures, find the maximum and minimum values of f(x,y) in the region representing the feasible solutions. (14)
1.f(x,y)xy2.f(x,y)2x3y
3.f(x,y)xy4.f(x,y)2yx
Find the maximum and minimum values of f(x,y) subject to each of the following systems of inequalities. (512)
5.6.
7.8.
9./ 10.
11.12.
Level 2
13.A small car park provides monthly parking for xprivate cars and y motor cycles. The coordinates of the points in the shaded region as shown in the figure satisfy all the constraints about x and y. If the profit of providing parking space for a private car is $500 per month and that for a motor cycle is $400 per month, how many private cars and motor cycles should be parked in the car park in order to obtain the maximum profit?
14.(a)Solve the following system of inequalities graphically.