1. 9: Linear and Quadratic Inequalities © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

2. Linear and Quadratic Inequalities Module C1 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

3. Linear and Quadratic Inequalities  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 by  1 gives  1 and  2 BUT  1 is greater than  2 So, We know ( 1 is less than 2 ) 11 22 

4. Linear and Quadratic Inequalities 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. Dividing by  1 gives  1 and  2 BUT  1 is greater than  2 So, The inequality has been reversed We know ( 1 is less than 2 ) Consider the numbers 1 and 2 :

5. Linear and Quadratic Inequalities e.g.1. Find the values of that satisfy the inequality Divide by  4 : Solution: Divide by 3 e.g.2 Find the range of values of x that satisfy the inequality Solution: Collect the like terms Notice the change from “less than” to “greater than”  Collecting the x -terms on the side which makes the coefficient positive avoids the need to divide by a negative number  Substitute one value of x as a check on the answer Tips:

6. Linear and Quadratic Inequalities Exercises Find the range of values of x satisfying the following linear inequalities: 1. 2. Solution: Solution: Either OrDivide by -4: so,

7. Linear and Quadratic Inequalities 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 find the zeros of Method: ALWAYS use a sketch

8. Linear and Quadratic Inequalities 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

9. Linear and Quadratic Inequalities 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”.

10. Linear and Quadratic Inequalities  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

11. Linear and Quadratic Inequalities 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:

12. Linear and Quadratic Inequalities

13. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

14. Linear and Quadratic Inequalities  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

15. Linear and Quadratic Inequalities e.g.1. Find the values of that satisfy the inequality Divide by -4 : Solution: Divide by 3 e.g.2 Find the range of values of x that satisfy the inequality Solution: Collect the like terms Notice the change from “less than” to “greater than”  Collecting the x -terms on the side which makes the coefficient positive avoids the need to divide by a negative number  Substitute one value of x as a check on the answer Tips: Linear Inequalities

16. Linear and Quadratic Inequalities 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 find the zeros of Method: ALWAYS use a sketch

17. Linear and Quadratic Inequalities 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