1. Chapter 11 Inequalities and Linear Programming

2. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Concepts of Inequalities In our daily life, people come across many problems involving concepts of inequalities.

3. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Speed Limits They indicate the speed limit for vehicles on the road. Anyone driving over the limit will break the law. What is the meaning of the numbers in the following road signs?

4. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming What is the restriction of vehicles indicated in the road sign? The height of vehicles cannot exceed 4.5 m. Restriction of vehicles

5. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Restriction of vehicles What is the restriction of vehicles indicated in the road sign? The length of vehicles cannot exceed 10 m.

6. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Restriction of vehicles What is the restriction of vehicles indicated in the road sign? The width of vehicles cannot exceed 2.3 m.

7. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Other Restrictions Give some examples in our daily life which involve concepts of inequalities, and explain their restrictions.

8. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Inequality Signs In mathematics, we may use an inequality sign to compare two unequal quantities. Inequality sign MeaningExample  less than x  2  less than or equal to; not greater than; at most y  10  greater than a  2  greater than or equal to; not less than; at least b  4  not equal to p  5

9. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Graphical Representations of Inequalities e.g. x  2 The symbol “  ” indicates that the number corresponds to that position is not a solution. 2

10. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Graphical Representations of Inequalities e.g. y  10 The symbol “  ” indicates that the number corresponds to that position is a solution. 10

11. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Graphical Representations of Inequalities e.g. a  7 The symbol “  ” indicates that the number corresponds to that position is not a solution. 7

12. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Graphical Representations of Inequalities e.g. b  4 The symbol “  ” indicates that the number corresponds to that position is a solution. 4

13. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Graphical Representations of Inequalities e.g. p  5 5

14. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Simultaneous Inequalities (I) 2 7 2 7 x  7

15. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Simultaneous Inequalities (II) 2 7 2 7 x  2x  2

16. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming 2 7 2 7 2  x  7 Simultaneous Inequalities (III)

17. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Simultaneous Inequalities (IV) 2 7 2 7 No solution

18. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Linear Inequalities in Two Variables x  2y  4

19. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Linear Inequalities in Two Variables x  2y  4 A dotted line

20. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Linear Inequalities in Two Variables x  2y  4 A solid line

21. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Linear Inequalities in Two Variables x  2y  4 A dotted line

22. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Linear Inequalities in Two Variables x  2y  4 A solid line

23. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming System of Linear Inequalities (I) x  2y  4 2x  y  42x  y  4 Region of feasible solutions

24. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming System of Linear Inequalities (II) x  y  5 2x  y  4 x  3y  3 Region of feasible solutions

25. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Linear Programming  George Dantzig (1914 - ), an American mathematician, invented linear programming in 1947.  He first applied linear programming in solving the problem of how to get adequate nutrition of the body at the minimum cost.

26. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Linear Programming In the figure, the yellow region represents the region of feasible solutions under certain constraints. What is the maximum value of f(x, y) = 2x  y in this region?

27. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Linear Programming f(2, 5)  2(2)  (5)  9 f(x, y) = 2x  y

28. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming Linear Programming f(5, 8)  2(5)  (8)  18 f(x, y) = 2x  y

29. 2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 11 Inequalities and Linear Programming f(12, 6)  2(12)  (6)  30 Linear Programming f(x, y) = 2x  y  When x  12 and y  6, f(x, y)  2x  y has the maximum value in this region.

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