1. HKDSE Mathematics Ronald Hui Tak Sun Secondary School

2. Missing Homework  Summer Holiday Homework 1  1, 10, 14  Summer Holiday Homework 2  1, 9, 14  SHW1-R1  9, 10 22 October 2015 Ronald HUI

3. Missing Homework  SHW2-A1  10  SHW2-B1  9  SHW2-C1  8, 9, 10, 12, 13, 14, 20  RE2  9 22 October 2015 Ronald HUI

4. Missing Homework  SHW2-R1  I got 8 only!  SHW2-P1  I got 4 only!!!  SHW3-01, 3-A1, 3-B1  Today!!!! 22 October 2015 Ronald HUI

5. Summary on “AND” 22 October 2015 Ronald HUI

6. Summary on “OR” 22 October 2015 Ronald HUI

7. Book 5A Chapter 3 Solving Quadratic Inequalities in One Unknown by the Graphical Method

8. If an inequality contains only one unknown and the highest degree of the terms is 2, the inequality is called a quadratic inequality in one unknown. Example: (i) x 2 + 3x – 10 > 0 (ii) x 2 – 2x + 1  0 (iv) –2x 2 + x + 1  0 (iii) x 2 + 3x – 10 < 0

9. We have to find all values of x satisfying the quadratic inequality. How can we solve this kind of quadratic inequalities?

10. Let’s see how to solve x 2 + 2x – 3 > 0 by the graphical method.

11. Solve the quadratic inequality x 2 + 2x – 3 > 0 graphically. Consider the graph of the quadratic function y = x 2 + 2x – 3. Points above the x-axis Points below the x-axis y > 0 y < 0

12. Solve the quadratic inequality x 2 + 2x – 3 > 0 graphically. Consider the graph of the quadratic function y = x 2 + 2x – 3. For x < –3: The graph is above the x-axis. For –3 < x < 1: The graph is below the x-axis. For x > 1: The graph is above the x-axis. y > 0 y < 0 y > 0

13. For y > 0 (i.e. x 2 + 2x – 3 > 0), ∴ The solutions of x 2 + 2x – 3 > 0 are x 1. For x < –3: y > 0 For –3 < x < 1: y < 0 For x > 1: y > 0 Solve the quadratic inequality x 2 + 2x – 3 > 0 graphically. Consider the graph of the quadratic function y = x 2 + 2x – 3. For y > 0 (i.e. x 2 + 2x – 3 > 0), the range of values of x are x 1.

14. Solve the quadratic inequality x 2 + 2x – 3 > 0 graphically. Consider the graph of the quadratic function y = x 2 + 2x – 3. Using the above graph, we can also find the solutions of x 2 + 2x – 3  0. For x 0 For –3 < x < 1: y < 0 For x > 1: y > 0

15. Solve the quadratic inequality x 2 + 2x – 3  0 graphically. For x  –3: The graph is on or above the x-axis. y  0 For –3  x  1: The graph is on or below the x-axis. y  0 For x  1: The graph is on or above the x-axis. y  0

16. the range of values of x are –3  x  1. Solve the quadratic inequality x 2 + 2x – 3  0 graphically. Note: In fact, the inequalities x 2 + 2x – 3  0 and x 2 + 2x – 3 < 0 can also be solved using the graph above. ∴ The solutions of x 2 + 2x – 3  0 are –3  x  1. For y  0 (i.e. x 2 + 2x – 3  0), For x  –3: y  0 For –3  x  1: y  0 For x  1: y  0

17. Let’s see how we can solve the quadratic inequality x 2 + 3x – 10 < 0 without any given graph.

18. Consider the corresponding quadratic function Solve the quadratic inequality x 2 + 3x – 10 < 0 graphically. Step 1 Write down the corresponding quadratic function y = ax 2 + bx + c. y = x 2 + 3x – 10.

19. Step 2 Find the x-intercept(s) and the direction of opening of the graph of the quadratic function. Then, sketch the graph. Solve the quadratic inequality x 2 + 3x – 10 < 0 graphically. Sketch the graph of y = x 2 + 3x – 10. x 2 + 3x – 10 = 0 (x + 5)(x – 2) = 0 x = –5 or x = 2 When y = 0,  ∴ The x-intercepts of the graph are –5 and 2.

20. Step 2 Find the x-intercept(s) and the direction of opening of the graph of the quadratic function. Then, sketch the graph. Solve the quadratic inequality x 2 + 3x – 10 < 0 graphically. Sketch the graph of y = x 2 + 3x – 10.  ∵ The coefficient of x 2 is 1 (> 0). ∴ The graph opens upwards. y = x 2 + 3x – 10

21. Read the solutions of the quadratic inequality from the graph. Step 3 y = x 2 + 3x – 10 For y < 0 (i.e. x 2 + 3x – 10 < 0), the corresponding part of the graph is below the x-axis. Solve the quadratic inequality x 2 + 3x – 10 < 0 graphically.

22. y = x 2 + 3x – 10 ∵ The graph of y = x 2 + 3x – 10 is below the x-axis when –5 < x < 2. ∴ The solutions of x 2 + 3x – 10 < 0 are –5 < x < 2. Solve the quadratic inequality x 2 + 3x – 10 < 0 graphically. Read the solutions of the quadratic inequality from the graph. Step 3

23. Follow-up question Solve the quadratic inequality –2x 2 + x + 1  0 graphically. Consider the corresponding quadratic function y = –2x 2 + x + 1. When y = 0, –2x 2 + x + 1 = 0 2x 2 – x – 1 = 0 (2x + 1)(x – 1) = 0 2 1 x = – or x = 1 ∴ The x-intercepts of the graph of y = –2x 2 + x + 1 are – and 1. 2 1

24. ∵ The coefficient of x 2 is –2 (< 0). ∴ The graph of y = –2x 2 + x + 1 opens downwards. Sketch the graph of y = –2x 2 + x + 1: From the graph, the solutions of –2x 2 + x + 1  0 are x  – or x  1. 2 1