# 11 x1 t03 01 inequations & inequalities (2012)

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<ul><li> 1. Inequations &amp; Inequalities</li></ul> <p> 2. Inequations &amp; Inequalities1. Quadratic Inequations e.g. (i ) x 2 5 x 6 0 3. Inequations &amp; Inequalities1. Quadratic Inequations e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 4. Inequations &amp; Inequalities1. Quadratic Inequationsy e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 1 x 5. Inequations &amp; Inequalities1. Quadratic Inequationsy e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 1 x Q: for what values of x is theparabola below the x axis? 6. Inequations &amp; Inequalities1. Quadratic Inequationsy e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 1 x Q: for what values of x is theparabola below the x axis? 7. Inequations &amp; Inequalities1. Quadratic Inequationsy e.g. (i ) x 2 5 x 6 0 x 6 x 1 06 x 1 6 1 x Q: for what values of x is theparabola below the x axis? 8. Inequations &amp; Inequalities1. Quadratic Inequationsy e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 x 16 1 x Q: for what values of x is the (ii ) x 2 3 x 4 0parabola below the x axis? 9. Inequations &amp; Inequalities1. Quadratic Inequationsy e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 x 16 1 x Q: for what values of x is the (ii ) x 2 3 x 4 0parabola below the x axis? x 2 3x 4 0 10. Inequations &amp; Inequalities1. Quadratic Inequationsy e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 x 1 61xQ: for what values of x is the (ii ) x 2 3 x 4 0 parabola below the x axis?x 2 3x 4 0y x 4 x 1 0 4 1x 11. Inequations &amp; Inequalities1. Quadratic Inequations y e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 x 1 6 1 xQ: for what values of x is the (ii ) x 2 3 x 4 0 parabola below the x axis?x 2 3x 4 0 y x 4 x 1 041x Q: for what values of x is theparabola above the x axis? 12. Inequations &amp; Inequalities1. Quadratic Inequations y e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 x 1 6 1 xQ: for what values of x is the (ii ) x 2 3 x 4 0 parabola below the x axis?x 2 3x 4 0 y x 4 x 1 041x Q: for what values of x is theparabola above the x axis? 13. Inequations &amp; Inequalities1. Quadratic Inequationsy e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 x 16 1 x Q: for what values of x is the (ii ) x 2 3 x 4 0parabola below the x axis?x 2 3x 4 0y x 4 x 1 0x 4 or x 1 Note: quadratic inequalities41xalways have solutions in the form Q: for what values of x is the? &lt; x &lt; ? OR x &lt; ? or x &gt; ?parabola above the x axis? 14. Inequations &amp; Inequalities1. Quadratic Inequations y e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 x 1 6 1 xQ: for what values of x is the (ii ) x 2 3 x 4 0 parabola below the x axis?x 2 3x 4 0 y x 4 x 1 0x 4 or x 141x Q: for what values of x is theparabola above the x axis? 15. Inequations &amp; Inequalities1. Quadratic Inequationsy e.g. (i ) x 2 5 x 6 0 x 6 x 1 0 6 x 16 1 x Q: for what values of x is the (ii ) x 2 3 x 4 0parabola below the x axis?x 2 3x 4 0y x 4 x 1 0x 4 or x 1 Note: quadratic inequalities41xalways have solutions in the form Q: for what values of x is the? &lt; x &lt; ? OR x &lt; ? or x &gt; ?parabola above the x axis? 16. 2. Inequalities with Pronumerals in the Denominator 17. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3x 18. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zerox 19. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 20. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 2) Solve the equality 21. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x1 2) Solve the equality 3 x 1 x3 22. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line 23. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line01 3 24. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line01 4) Test regions 3 25. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line01 4) Test regions 3 1 Test x 1 31 26. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line01 4) Test regions 3 1 1 1 3 Test x 1 3 Test x 11 4 4 27. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line01 4) Test regions 3 1 1 1 3 Test x 1 3 Test x 11 4 4 28. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line 01 4) Test regions3 1 1 1 31 Test x 1 3 Test x 1Test x 1 31 4 1 4 29. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line 01 4) Test regions3 1 1 1 31 Test x 1 3 Test x 1Test x 1 31 4 1 4 10 x 3 30. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line 01 4) Test regions3 111 31 Test x 1 3 Test x 1Test x 1 31 4 1 4 10 x 2 3(ii ) 5x3 31. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line 01 4) Test regions3 1 1 1 31 Test x 1 3 Test x 1Test x 1 31 4 1 4 10 x 2 3(ii ) 5x3x3 0x 3 32. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x 1 2) Solve the equality3x 1 x3 3) Plot these values on a number line 01 4) Test regions3 1 1 1 31 Test x 1 3 Test x 1Test x 1 31 4 1 4 10 x 2 3(ii ) 52x3 5x3x3 0x 3 33. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x1 2) Solve the equality 3 x 1 x3 3) Plot these values on a number line 01 4) Test regions3 1 1 1 31 Test x 1 3Test x 1 Test x 1 31 4 1 4 1 0 x 23(ii ) 52x35x3x3 0 2 5 x 15x 35 x 1313x 5 34. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x1 2) Solve the equality 3 x 1 x3 3) Plot these values on a number line 0 1 4) Test regions 3 1 1 1 3 1 Test x 1 3Test x 1 Test x 131 41 4 1 0 x 23(ii ) 52x35x3x3 0 2 5 x 15313 x 35 x 13 513x 5 35. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x1 2) Solve the equality 3 x 1 x3 3) Plot these values on a number line 0 1 4) Test regions 3 1 1 1 3 1 Test x 1 3Test x 1 Test x 131 41 4 1 0 x 23(ii ) 52x35x3x3 0 2 5 x 15313 x 35 x 13 513x 5 36. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x1 2) Solve the equality 3 x 1 x3 3) Plot these values on a number line 0 1 4) Test regions 3 1 1 1 3 1 Test x 1 3Test x 1 Test x 131 41 4 1 0 x 23(ii ) 52x35x3x3 0 2 5 x 15313 x 35 x 13 513x 5 37. 2. Inequalities with Pronumerals in the Denominator1 e.g. (i ) 3 1) Find the value where the denominator is zero x 0x1 2) Solve the equality 3 x 1 x3 3) Plot these values on a number line01 4) Test regions 3 1 1 1 31 Test x 1 3Test x 1Test x 131 4 1 4 1 0 x 23(ii ) 52x35x3x3 0 2 5 x 15 313x 35 x 135 13x13 x 3 or x 55 38. 3. Proving Inequalities 39. 3. Proving Inequalities(I) Start with a known result 40. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2 41. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2x y yz 42. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2x y yz 2 y z x 43. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2x y yz 2 y z xxzy2 44. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2x y yz 2 y z xxzy2(II) Move everything to the left 45. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2x y yz 2 y z xxzy2(II) Move everything to the left Show that if a 0, b 0 then ab a 2 b 2 2a 2b 2 46. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2x y yz 2 y z xxzy2(II) Move everything to the left Show that if a 0, b 0 then ab a 2 b 2 2a 2b 2 ab a 2 b 2 2a 2b 2 47. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2x y yz 2 y z xxzy2(II) Move everything to the left Show that if a 0, b 0 then ab a 2 b 2 2a 2b 2 ab a 2 b 2 2a 2b 2 ab a 2 2ab b 2 48. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2x y yz 2 y z xxzy2(II) Move everything to the left Show that if a 0, b 0 then ab a 2 b 2 2a 2b 2 ab a 2 b 2 2a 2b 2 ab a 2 2ab b 2 ab a b 2 49. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2x y yz 2 y z xxzy2(II) Move everything to the left Show that if a 0, b 0 then ab a 2 b 2 2a 2b 2 ab a 2 b 2 2a 2b 2 ab a 2 2ab b 2 ab a b 20 50. 3. Proving Inequalities(I) Start with a known resultxz If x y y z prove y 2x y yz 2 y z xxzy2(II) Move everything to the left Show that if a 0, b 0 then ab a 2 b 2 2a 2b 2 ab a 2 b 2 2a 2b 2 ab a 2 2ab b 2 ab a b 20 ab a 2 b 2 2a 2b 2 51. (III) Squares are positive or zero 52. (III) Squares are positive or zero Show that if a, b and c are positive, then a 2 b 2 c 2 ab bc ac 53. (III) Squares are positive or zero Show that if a, b and c are positive, then a 2 b 2 c 2 ab bc aca b 0 2 54. (III) Squares are positive or zero Show that if a, b and c are positive, then a 2 b 2 c 2 ab bc ac a b 0 2a 2 2ab b 2 0 55. (III) Squares are positive or zero Show that if a, b and c are positive, then a 2 b 2 c 2 ab bc ac a b 02a 2 2ab b 2 0a 2 b 2 2ab 56. (III) Squares are positive or zero Show that if a, b and c are positive, then a 2 b 2 c 2 ab bc ac a b 02a 2 2ab b 2 0a 2 b 2 2aba 2 c 2 2acb 2 c 2 2bc 57. (III) Squares are positive or zero Show that if a, b and c are positive, then a 2 b 2 c 2 ab bc ac a b 02a 2 2ab b 2 0a 2 b 2 2aba 2 c 2 2acb 2 c 2 2bc2a 2 2b 2 2c 2 2ab 2bc 2ac 58. (III) Squares are positive or zero Show that if a, b and c are positive, then a 2 b 2 c 2 ab bc ac a b 02a 2 2ab b 2 0a 2 b 2 2aba 2 c 2 2acb 2 c 2 2bc2a 2 2b 2 2c 2 2ab 2bc 2ac a 2 b 2 c 2 ab bc ac 59. (III) Squares are positive or zero Show that if a, b and c are positive, then a 2 b 2 c 2 ab bc ac a b 02a 2 2ab b 2 0a 2 b 2 2aba 2 c 2 2acb 2 c 2 2bc2a 2 2b 2 2c 2 2ab 2bc 2ac a 2 b 2 c 2 ab bc ac Exercise 3A; 4, 6ace, 7bdf, 8bdf, 9, 11, 12, 13ac, 15, 18bcd, 22, 24</p>