Ch-3 Lecture Notes on Inequalities and Identities.pptx

Inequilities and identities

1. Further Pure Mathematics Lecture 6 3 rd Batch PF101M

2. CHAPTER 3 Inequalities and Identities

3. Learning Objectives Solve Two Simultaneous Equations, One Linear and One Quadratic Solve Linear Inequalities

4. Starter Activities Using substitution, solve these simultaneous equations. Substitute in (2) Therefore Substitute in (1),  

5. Substitute in (2) Therefore Substitute in (1),   Starter Activities

6. Solve Solution So Solutions are   Example 1 Solve Two Simultaneous Equations, One Linear and One Quadratic

7. (b) Solve Solution Substitute x values into Solutions are  

8. Exercise 1

9. If the graphs of the lines in this system of equations intersect at (-3, 1), what is the value of    ?

10. Solve Linear Inequalities Example 2 Find the set of values of x for which Solution Solution   Solution  

12. Find the set of values of x for which Draw a number line illustrate the two inequalities. So the required set of values is   Example 3 -4 -2 2 4 6 8    

13. Draw a number line So the required set of values is   Example 4 Find the set of values of x for which -4 -2 2 4 6 8    

14. Exercise 2

15. Learning Objectives Solve Quadratic Inequalities Graph Linear Inequalities in Two Variables Divide a Polynomial by  

16. Solve Quadratic Inequalities To solve quadratic inequalities you need to Solve the quadratic equation Sketch the corresponding quadratic function Use your sketch to find the required set of values .

17. Find the set of values of x for which and draw a sketch to show this. Solution (critical values) The graph crosses the x-axis at -1 and 5. . y-intersect point is . So required set of values is   Example 5 y x  

18. Find the set of values of x for which and sketch the graph Solution (critical values) The graph crosses the x-axis at -3 and 1/2. y-intersect point is . So required set of values is   Example 6 y x  

19. Find the set of values of x for which Method 1. Sketch graph or Sketch of Therefore   Example 7   y x

20. Method 2. Table or is negative for Therefore                    

21. Exercise 3 By sketching graphs, solve these inequalities. (c) (d) 2. Find the set of values of x which satisfy these inequalities. (c) (f) (h) (k)  

22. Graph Linear Inequalities in Two Variables Consider the two diagrams below.                                 All the points that lie on the line are represented by All the points that lie below the line , in region B, are represented by All the points that lie above the line , in region A, are represented by  

23. When illustrating an equality, first draw a straight line and then use shading. Inequalities using or are represented by a dashed line . This indicates that the line itself is not included in the region. Inequalities using are represented by a solid line to show that the line is included in the region . The convention is that the region that does not satisfy the inequality is shaded. The feasible region satisfying several inequalities will be the one left unshaded.   This is the feasible region for these three inequalities.      

24. Write down the inequalities shown by regions A, B, C, D, E, F,G, H, I and J in the diagrams below. Example 8

25. Example 9 Illustrate on a diagram the region R, for which Label the region R.   keep     reject     Draw       Draw as a broken line. To draw it, note that when When   keep   reject                    

26.               keep   reject     To draw the line It pass through (0, 0) and (2, 1)         reject reject Keep     Combine these on one diagram:  

27. A food cart manager sells cheeseburgers and chicken burgers. The cart is open only during the lunch break of 12-2 pm. The manager can cook a maximum of 40 cheeseburgers and a maximum of 70 chicken burgers. He cannot cook more than 90 burgers in total. The profit on a cheeseburgers is 33 pence and the profit on a chicken burger is 21 pence. How many of each kind of burger should the manager sell to maximize profit? Example 10

28. Step 1. Formulate the problem into linear equations. Let x=number of cheeseburgers Let y= number of chicken burgers and you cannot have negative number of burgers Number of each burger and Total number of burgers Step 2. Obtain a profit inequality (convert pence to pounds) Step 3. Now draw these equations on a graph.  

29. Step 4 Find the coordinates of the vertices The coordinates of A are (0, 70) The coordinates of B are (20, 70) The coordinates of C are (40, 50) The coordinates of D are (40, 0)                         No shaded region satisfies all the inequalities. Maximum profit value will be closed to one of the vertices of the regions marked A, B, C, D.

30. Step 5 Input these coordinates into profit inequality founded in Step 2 For A For B For C For D Therefore, the maximum possible profit is £23.70 For this, the manager will need to sell 40 cheeseburgers and 50 chicken burgers .   A (0, 70), B(20, 70), C(40, 50), D(40, 0)  

31. Exercise 4 3-7

32. Divide a Polynomial by   A quotient in mathematics is the result of the division of two numbers. Example 11 Divide So                   5   5                

33. Divide   Example 12                 2   2                 So             The quotient is   The remainder is  

34. Exercise 5 (HW) Divide (d) (f) (h) 2. Divide (d)  

35. Learning Objectives Factor a Polynomial by Using the Factor Theorem Using the Remainder Theorem , Find the Remainder when a Polynomial is Divided by  

36. Factor a Polynomial by Using the Factor Theorem If is a polynomial and then x is a factor of  

37. Example 13 Show that is a factor of by   (a) Algebraic division (b) Factor theorem                 3   3                 (a) (b)         So is a factor of  

38. Example 14 F         5                                            

39. Example 15 G               Solution Example 16 G   is a factor of f(x) then   Solution   Where is a polynomial. So         So as required.  

40. Exercise 6 (HW) Use the factor theorem to show that is a factor of (c) ( is a factor of (e) ( is a factor of 2. Show that is exactly divisible by but not by 4. Show that is a factor of and factorize 7. Given that is a factor of Find the value of b. Hence factorize completely. 9. Given that and are factors of Find the values of a and b. factorize f(x) completely.  

41. By using the remainder theorem, you can find the remainder when a polynomial is divided by If polynomial f(x) is divided by then the remainder is A remainder in mathematics is what is left over in a division problem .   Using the Remainder Theorem , Find the Remainder when a Polynomial is Divided by  

42. Example 17 Find the remainder when is divided by using (a) Algebraic division (b) The remainder theorem             4                         The remainder is           The remainder is  

43. Example 18 When is divided by the remainder is 3. Find the value of a.                

44. Exercise 7 (HW) 1. In each of these equations, find the remainder using the remainder theorem. is divided by (d) divided by 2. In each of these equations, find the remainder using the algebraic division. is divided by (c) is divided by 4. When is divided by the remainder is 17. Find b. 7. When is divided by the remainder is 5 and when it is divided by the remainder is 6. Find the values of a and b . 10. The expression is divisible by but leave a remainder of 10 when divided by the values of p and q.  

45. Exam Practice for Chapter 3