1. Linear Inequalities Foundation Part I

2. An INEQUALITY shows a relationship between two variables, usually x & y Examples –y > 2x + 1 –y < x – 3 –3x 2 + 4y ≥ 12 What is an inequality?

3. Objective of these next few slides Read a graph and write down the inequalities that contain a region Draw inequalities and indicate the region they describe You need to know how to plot straight line graphs (yesterday)

4. For example x y x > 2 X=2 When dealing with ONE inequality, we SHADE IN the REQUIRED REGION

5. For example x y x < -2 X=-2

6. For example x y y < -1 y=-1

7. For example x y y < 2x +1 y= 2x+1 Which side is shaded? Pick a point NOT on line (1,2) Is 2 < 2 x 1 + 1 ? YES (1,2) lies in the required region

8. For example x y y > 3x - 2 y= 3x-2 Which side is shaded? Pick a point NOT on line (2,1) Is 1 > 3 x 2 - 2 ? NO (2,1) does NOT lie in the required region

9. How to draw graph of equation x y y = 3x + 2 Shade IN the Region for y > 3x + 2 (2,1) Is 1 > 3 x 2 + 2 ? NO (2,1) does NOT lie in the required region

10. How to draw graph of equation x y 4y + 3x = 12 Shade IN the Region for 4y + 3x > 12 (3,2) Is 4 x 2 + 3 x 3 > 12 ? YES (3,2) DOES lie in the required region

11. Regions enclosed by inequalities

12. y = 3 x = 4 x + y = 4 y < 3 x < 4 x + y > 4 (3,2) 2 < 3 ? 3 + 2 > 4 ? 3 < 4 ?

13. Part II Solving Linear and Quadratic Inequalities

14.  Linear Inequalities These inequalities can be solved like linear equations EXCEPT that multiplying or dividing by a negative number reverses the inequality. Consider the numbers 1 and 2 : Examples of linear inequalities: 1.2. Dividing or multiplying by  1 gives  1 and  2 BUT  1 is greater than  2 So, We know ( 1 is less than 2 ) 11 22 

15. Linear Inequalities These inequalities can be solved like linear equations EXCEPT that multiplying or dividing by a negative number reverses the inequality. Examples of linear inequalities: 1.2.

16. Exercises Find the range of values of x satisfying the following linear inequalities: 1. 2. Solution: Solution: Either OrDivide by -4: so,

17. Quadratic Inequalities Solution: e.g.1 Find the range of values of x that satisfy Rearrange to get zero on one side: or is less than 0 below the x -axis The corresponding x values are between -3 and 1 Let and solve Method: ALWAYS use a sketch

18. Solution: e.g.2 Find the values of x that satisfy or There are 2 sets of values of x Find the zeros of where is greater than or equal to 0 above the x -axis or These represent 2 separate intervals and CANNOT be combined

19. Solution: e.g.3 Find the values of x that satisfy Find the zeros of where is greater than 0 above the x -axis This quadratic has a common factor, x or Be careful sketching this quadratic as the coefficient of is negative. The quadratic is “upside down”.

20.  Linear inequalities Solve as for linear equations BUT Keep the inequality sign throughout the working If multiplying or dividing by a negative number, reverse the inequality  Quadratic ( or other ) Inequalities rearrange to get zero on one side, find the zeros and sketch the function Use the sketch to find the x -values satisfying the inequality Don’t attempt to combine inequalities that describe 2 or more separate intervals SUMMARY

21. Exercise or There are 2 sets of values of x which cannot be combined is greater than or equal to 0 above the x -axis or 1. Find the values of x that satisfy where Solution:

22. Now do Exercise 4A page 126