Chapter 11 Inequalities and Linear Programming Additional Example 11.1Additional Example 11.1 Additional Example 11.2Additional Example 11.2 Additional Example 11.3Additional Example 11.3 Additional Example 11.4Additional Example 11.4 Additional Example 11.5Additional Example 11.5 Additional Example 11.6Additional Example 11.6 Additional Example 11.7Additional Example 11.7 Additional Example 11.8Additional Example 11.8 Additional Example 11.9Additional Example 11.9 Additional Example 11.10Additional Example Example 1Example 1 Example 2Example 2 Example 3Example 3 Example 4Example 4 Example 5Example 5 Example 6Example 6 Example 7Example 7 Example 8Example 8 Example 9Example 9 Example 10Example 10 New Trend Mathematics - S4B Quit

Chapter 11 Inequalities and Linear Programming Additional Example 11.11Additional Example Additional Example 11.12Additional Example Additional Example 11.13Additional Example Additional Example 11.14Additional Example Additional Example 11.15Additional Example Additional Example 11.16Additional Example Example 11Example 11 Example 12Example 12 Example 13Example 13 Example 14Example 14 Example 15Example 15 Example 16Example 16 New Trend Mathematics - S4B Quit

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.1 Solve the following inequalities and represent the solutions graphically. Solution: Graphical representation:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.1 Graphical representation: Solution:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.2 Find the least values of three consecutive odd numbers such that 4 times the smallest number is not less than 3 times the largest one. Solution: Let x be the smallest odd number. Then the other two consecutive odd numbers are x  2 and x  4. Since x is an odd number, the least value of x is 13.

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.3 Solve the following simultaneous inequalities and represent the solutions graphically. Solution: Graphical representation of the solutions:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.3 Graphical representation of the solutions: Solution:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.4 Solve the following simultaneous inequalities and represent the solutions graphically. Solution: Graphical representation of the solutions:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.4 Graphical representation of the solutions: and Solution:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.5 Solve the following simultaneous inequalities and represent the solutions graphically. Solution: and Graphical representation of the solutions:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.5 Graphical representation of the solutions: and Solution:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.6 Solve the following simultaneous inequalities and represent the solutions graphically. Solution:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.6 and Solution:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Solution: and Graphical representation of the solutions: Additional Example 11.7 Solve the following simultaneous inequalities and represent the solutions graphically.

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.7 Solution: Graphical representation of the solutions: and

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.8 Solve the inequality 2x  3y  15 < 0 graphically. Solution: The graphical solution of 2x  3y  15 < 0 is shown below.

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example 11.9 Solve the system of inequalities graphically. Solution: The solutions of the system of inequalities are indicated by the shaded region in the figure.

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example Solve the system of inequalities graphically. Solution: The solutions of the system of inequalities are indicated by the shaded region in the figure.

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example Solve the following system of inequalities graphically. Solution: Since x and y are integers, only the coordinates of the points represented by the dots in the figure are the solutions.

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example There are x standard rooms and y deluxe rooms on a floor of a hotel. The floor area of each standard room is 20 m 2 and that of each deluxe room is 30 m 2. In order to satisfy the demand, the number of standard rooms should not be more than 2 times that of deluxe rooms, and the total number of rooms on that floor should not exceed 50. It is known that and the total floor area of the rooms is at most m 2. Write down all the constraints about x and y. Solution: The constraints are which are equivalent to

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example Martin wants to prepare fruit juice A and B for a party with 4 L of pineapple juice and 6 L of orange juice. Fruit juice A and B are prepared by mixing pineapple juice and orange juice in the ratios of 1 : 1 and 3 : 5 respectively. It is known that the amount of fruit juice B prepared should be at least twice that of fruit juice A. Let x L and y L be the amounts of fruit juice A and B prepared respectively. (a)Write down all the constraints about x and y. (b)Find the feasible solutions graphically.

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Solution: (a)The constraints are which are equivalent to Additional Example 11.13

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit (b) The shaded region in the figure indicates all the feasible solutions. Solution: Additional Example 11.13

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example Find the maximum and minimum values of each of the following functions subject to the system of inequalities in Example 14. (a)f(x, y)  x  2y (b)g(x, y)  x  y Solution: (a)At (  1, 2), f (  1, 2)   1  2(2)  3. At (2, 1), f (2, 1)  2  2(1)  4. At (3, 5), f (3, 5)  3  2(5)  13.  (b)At (  1, 2), g (  1, 2)   1  2   3. At (2, 1), g (2, 1)  2  1  1. At (3, 5), g (3, 5)  3  5   2. 

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example A salt plant operates 16 hours a day to produce x bags of table salt and y bags of coarse salt. It is known that at least 900 bags of table salt and coarse salt should be produced everyday and the amount of coarse salt produced should be at least 3 times that of the table salt. To produce a bag of table salt and a bag of coarse salt, 60 seconds and 30 seconds are required respectively. (a)Write down all the constraints about x and y. (b)Draw and shade the region which satisfies all the constraints. (c)If the profit of producing a bag of table salt is 2.5 times as much as that of coarse salt, how many bags of table salt and coarse salt should be produced in order to maximize the profit?

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example Solution: which are equivalent to (a)The constraints are

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example (c)Let the profit of producing a bag of coarse salt be $k (where k > 0), then the profit of producing a bag of table salt is $2.5k. Therefore, total profit From the graph the total profit is at the maximum when x  384, y  (b) The feasible solutions are all integral solutions in the shaded region. Solution:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example A shop sells two different stationery sets A and B. The relevant data of the two sets are as follows: Mr. Chan needs to buy at least 150 notebooks, 100 correction pens and 240 pens with at most $ Let x and y be the respective number of packs of sets A and B to be bought. (a)Write down all the constraints about x and y. (b)Draw and shade the region which satisfies all the constraints. (c)What is the most economical way to buy the required stationery? Find the minimum cost.

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example Solution: which are equivalent to (a)The constraints are

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit Additional Example (b) The feasible solutions are all integral solutions in the shaded region. Solution:

Chapter 11 Inequalities and Linear Programming 2004 Chung Tai Educational Press © Quit (c)Cost c(x, y)  $(25x + 20y)  The minimum value is attained at B(25, 75).  Minimum cost Solution: Additional Example 11.16