1. 6.7 Graph and Solving Quadratic Inequalities

2. Method of Graph sketching

8. Solving Quadratic Inequalities

9. Solve the quadratic inequality x 2 – 5 x + 6 > 0 graphically. Example 1:

10. Procedures: Step (2): we have y = ( x – 2)( x – 3) , i.e. y = 0, when x = 2 or x = 3. Factorize x 2 – 5 x + 6, The corresponding quadratic function is y = x 2 – 5 x + 6 Sketch the graph of y = x 2 – 5 x + 6. Step (1): Step (3): Step (4): Find the solution from the graph.

11. Sketch the graph y = x 2 – 5 x + 6 . What is the solution of x 2 – 5 x + 6 > 0 ? x y 0 y = ( x – 2)( x – 3) ,  y = 0, when x = 2 or x = 3.  2 3

12. x y 0 We need to solve x 2 – 5 x + 6 > 0, The portion of the graph above the x-axis represents y > 0 (i.e. x 2 – 5 x + 6 > 0) The portion of the graph below the x-axis represents y < 0 (i.e. x 2 – 5 x + 6 < 0) above the x-axis. so we choose the portion 2 3

13. x y 0 When x < 2 , the curve is above the x-axis i.e., y > 0 x 2 – 5x + 6 > 0 When x > 3 , the curve is above the x-axis i.e., y > 0 x 2 – 5x + 6 > 0 2 3

14. From the sketch, we obtain the solution or

15. Graphical Solution: 0 2 3

16. Solve the quadratic inequality x 2 – 5 x + 6 < 0 graphically. Example 2: Same method as example 1 !!!

17. x y 0 When 2 < x < 3 , the curve is below the x-axis i.e., y < 0 x 2 – 5 x + 6 < 0 2 3

18. From the sketch, we obtain the solution 2 < x < 3

19. Graphical Solution: 0 2 3

22. Solve Exercise 3 : – 7 < x < 5 Solution: Find the x-intercepts of the curve: (x + 7)(x – 5)=0 x = –7 or x = 5 x y 0 0 – 7 5 – 7 5

23. Solve Exercise 4 : Solution: Find the x-intercepts of the curve: (x + 3)(3x – 2)=0 x = –3 or x = 2/3 x  –3 or x  2/3 x y 0 – 3 2 3 0 – 3 2 3