1. Year 9 Inequalities Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 23 rd March 2015 Objectives: Solving linear inequalities, combining inequalities and representing solutions on number lines.

2. Means: x is less than or equal to 4. Writing inequalities and drawing number lines You need to be able to sketch equalities and strict inequalities on a number line. x > 3 Means: x is (strictly) greater than 3. 0 1 2 3 4 5 ? This is known as a ‘strict’ inequality. x < -1 Means: x is (strictly) less than -1. -3 -2 -1 0 1 2 ? x ≥ 4 Means: x is greater than or equal to 4. 2 3 4 5 6 7 ? x ≤ 5 2 3 4 5 6 7 ? ? ? ? ?

3. Deal or No Deal? We can manipulate inequalities in various ways, but which of these are allowed and not allowed?  Click to DealClick to No Deal Can we add or subtract to both sides?

4. Deal or No Deal? We can manipulate inequalities in various ways, but which of these are allowed and not allowed?  Click to DealClick to No Deal Can we divide both sides by a positive number?

5. Deal or No Deal? We can manipulate inequalities in various ways, but which of these are allowed and not allowed?  Click to DealClick to No Deal Can we multiply both sides by a positive number?

6. Deal or No Deal? We can manipulate inequalities in various ways, but which of these are allowed and not allowed?  Click to DealClick to No Deal Can we multiply both sides by a negative number?

7. 2 If we multiply or divide both sides of the inequality by a negative number, the inequality ‘flips’! <4 Click to start Bro-manimation × (-1) -2 × (-1) -4 OMG magic! ‘Flipping’ the inequality

8. Alternative Approach Or you could simply avoid dividing by a negative number at all by moving the variable to the side that is positive. ? ? ? ? ? ?

9. Solve Quickfire Examples ? ? ? ? ?

10. Deal or No Deal? We can manipulate inequalities in various ways, but which of these are allowed and not allowed?  Click to DealClick to No Deal Can we multiply both sides by a variable? The problem is, we don’t know if the variable has a positive or negative value, so negative solutions would flip it and positive ones wouldn’t. You won’t have to solve questions like this until Further Maths A Level!

11. Solve Hint: Do the addition/subtraction before you do the multiplication/division. ? ? ? ? ? More Examples

12. 8 < 5x - 2 ≤ 23 Hint: Do the addition/subtraction before you do the multiplication/division. 8 < 5x - 25x - 2 ≤ 23 and 2 < x and x ≤ 5 2 < xx ≤ 5 Click to start bromanimation Dealing with multiple inequalities

13. Solve ? Hint: Do the addition/subtraction before you do the multiplication/division. Solve ? More Examples

14. Test Your Understanding Solve ? ?

15. Exercise 1 Solve the following inequalities, and illustrate each on a number line: 1 2 3 4 5 6 7 8 9 10 11 11 ? ? ? ? ? ? ? ? ? ? ? ? 22 ?

16. Combining inequalities It’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable. x ≥ 2 and x < 4 AND How would we express “x is greater than or equal to 2, and less than 4”? x ≥ 2, x < 4 2 ≤ x < 4 This last one emphasises the fact that x is between 2 and 4. ? ? ? OR How would we express “x is less than -1, or greater than 3”? x 3 ? This is the only way you would write this – you must use the word ‘or’.

17. Combining inequalities It’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable. 2 ≤ x < 4x 4 0 1 2 3 4 5 ? -1 0 1 2 3 4 ?

18. Combining inequalities It’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable. x ≥ 2 and x < 4x 4 0 1 2 3 4 5 ? -1 0 1 2 3 4 ? or and To illustrate the difference, what happens when we switch them?

19. I will shoot you if I see any of these… This is technically equivalent to: x > 7 This is technically equivalent to: x < 4 The least offensive of the three, but should be written: 4 < x < 7 ? ? ?

20. Combining Inequalities 25 4 25 4 Combined ? ? In general, we can combine inequalities either by common sense, or using number lines... Where are you on both lines?

21. Test Your Understanding 5 -3 Combined 3 ? ? ? ? 1 st condition 2 nd condition

22. Exercise 2 By sketching the number lines or otherwise, combine the following inequalities. 1 2 3 4 5 6 7 8 9 10 11 ? ? ? ? ? ? ? ? ? ? 12 13 14 15 ? ? ? ? ?