11x1 t03 01 inequations & inequalities (2011)
TRANSCRIPT
![Page 1: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/1.jpg)
Inequations & Inequalities
![Page 2: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/2.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
![Page 3: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/3.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
![Page 4: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/4.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
![Page 5: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/5.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
![Page 6: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/6.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
![Page 7: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/7.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
6 1x
![Page 8: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/8.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
6 1x
2( ) 3 4 0ii x x
![Page 9: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/9.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
6 1x
2( ) 3 4 0ii x x 2 3 4 0x x
![Page 10: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/10.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
6 1x
2( ) 3 4 0ii x x 2 3 4 0x x
4 1 0x x y
x–4 1
![Page 11: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/11.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
6 1x
2( ) 3 4 0ii x x 2 3 4 0x x
4 1 0x x y
x–4 1Q: for what values of x is the
parabola above the x axis?
![Page 12: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/12.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
6 1x
2( ) 3 4 0ii x x 2 3 4 0x x
4 1 0x x y
x–4 1Q: for what values of x is the
parabola above the x axis?
![Page 13: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/13.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
6 1x
2( ) 3 4 0ii x x 2 3 4 0x x
4 1 0x x y
x–4 1Q: for what values of x is the
parabola above the x axis?
4 or 1x x
Note: quadratic inequalities always have solutions in the form ? < x < ? OR x < ? or x > ?
![Page 14: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/14.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
6 1x
2( ) 3 4 0ii x x 2 3 4 0x x
4 1 0x x y
x–4 1Q: for what values of x is the
parabola above the x axis?
4 or 1x x
![Page 15: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/15.jpg)
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
Q: for what values of x is theparabola below the x axis?
6 1x
2( ) 3 4 0ii x x 2 3 4 0x x
4 1 0x x y
x–4 1Q: for what values of x is the
parabola above the x axis?
4 or 1x x
Note: quadratic inequalities always have solutions in the form ? < x < ? OR x < ? or x > ?
![Page 16: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/16.jpg)
2. Inequalities with Pronumerals in the Denominator
![Page 17: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/17.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix
![Page 18: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/18.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero
![Page 19: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/19.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
![Page 20: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/20.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”
![Page 21: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/21.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
![Page 22: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/22.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line
![Page 23: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/23.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0 1
3
![Page 24: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/24.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions13
![Page 25: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/25.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
13
![Page 26: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/26.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
13
![Page 27: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/27.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
13
![Page 28: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/28.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
Test 1x 1 31
13
![Page 29: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/29.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
Test 1x 1 31
13
103
x
![Page 30: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/30.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
Test 1x 1 31
13
103
x 2( ) 5
3ii
x
![Page 31: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/31.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
Test 1x 1 31
13
103
x 2( ) 5
3ii
x
3 0x
3x
![Page 32: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/32.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
Test 1x 1 31
13
103
x 2( ) 5
3ii
x
3 0x
3x
2 53x
![Page 33: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/33.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
Test 1x 1 31
13
103
x 2( ) 5
3ii
x
3 0x
3x
2 53x
2 5 15x
5 13x 135
x
![Page 34: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/34.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
Test 1x 1 31
13
103
x 2( ) 5
3ii
x
3 0x
3x
2 53x
2 5 15x
5 13x 135
x
–3 135
![Page 35: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/35.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
Test 1x 1 31
13
103
x 2( ) 5
3ii
x
3 0x
3x
2 53x
2 5 15x
5 13x 135
x
–3 135
![Page 36: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/36.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
Test 1x 1 31
13
103
x 2( ) 5
3ii
x
3 0x
3x
2 53x
2 5 15x
5 13x 135
x
–3 135
![Page 37: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/37.jpg)
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”1 3x
13
x
3) Plot these values on a number line0
4) Test regions
Test 1x 1 31
1Test 4
x 1 314
Test 1x 1 31
13
103
x 2( ) 5
3ii
x
3 0x
3x
2 53x
2 5 15x
5 13x 135
x
–3 135
133 or 5
x x
![Page 38: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/38.jpg)
3. Proving Inequalities
![Page 39: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/39.jpg)
3. Proving Inequalities(I) Start with a known result
![Page 40: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/40.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
![Page 41: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/41.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z
![Page 42: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/42.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z 2y z x
![Page 43: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/43.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z 2y z x
2x zy
![Page 44: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/44.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z 2y z x
2x zy
(II) Move everything to the left
![Page 45: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/45.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z 2y z x
2x zy
2 2 2 2Show that if 0, 0 then 2a b ab a b a b
(II) Move everything to the left
![Page 46: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/46.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z 2y z x
2x zy
2 2 2 2Show that if 0, 0 then 2a b ab a b a b
2 2 2 22ab a b a b
(II) Move everything to the left
![Page 47: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/47.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z 2y z x
2x zy
2 2 2 2Show that if 0, 0 then 2a b ab a b a b
2 2 2 22ab a b a b 2 22ab a ab b
(II) Move everything to the left
![Page 48: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/48.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z 2y z x
2x zy
2 2 2 2Show that if 0, 0 then 2a b ab a b a b
2 2 2 22ab a b a b 2 22ab a ab b
2ab a b
(II) Move everything to the left
![Page 49: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/49.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z 2y z x
2x zy
2 2 2 2Show that if 0, 0 then 2a b ab a b a b
2 2 2 22ab a b a b 2 22ab a ab b
2ab a b 0
(II) Move everything to the left
![Page 50: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/50.jpg)
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z 2y z x
2x zy
2 2 2 2Show that if 0, 0 then 2a b ab a b a b
2 2 2 22ab a b a b
2 2 2 22ab a b a b
2 22ab a ab b
2ab a b 0
(II) Move everything to the left
![Page 51: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/51.jpg)
(III) Squares are positive or zero
![Page 52: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/52.jpg)
(III) Squares are positive or zero2 2 2Show that if , and are positive, then a b c a b c ab bc ac
![Page 53: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/53.jpg)
(III) Squares are positive or zero2 2 2Show that if , and are positive, then a b c a b c ab bc ac
2 0a b
![Page 54: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/54.jpg)
(III) Squares are positive or zero2 2 2Show that if , and are positive, then a b c a b c ab bc ac
2 0a b 2 22 0a ab b
![Page 55: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/55.jpg)
(III) Squares are positive or zero2 2 2Show that if , and are positive, then a b c a b c ab bc ac
2 2 2a b ab
2 0a b 2 22 0a ab b
![Page 56: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/56.jpg)
(III) Squares are positive or zero2 2 2Show that if , and are positive, then a b c a b c ab bc ac
2 2 2a b ab
2 0a b 2 22 0a ab b
2 2 2a c ac 2 2 2b c bc
![Page 57: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/57.jpg)
(III) Squares are positive or zero2 2 2Show that if , and are positive, then a b c a b c ab bc ac
2 2 2a b ab
2 0a b 2 22 0a ab b
2 2 2a c ac 2 2 2b c bc
2 2 22 2 2 2 2 2a b c ab bc ac
![Page 58: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/58.jpg)
(III) Squares are positive or zero2 2 2Show that if , and are positive, then a b c a b c ab bc ac
2 2 2a b ab
2 0a b 2 22 0a ab b
2 2 2a c ac 2 2 2b c bc
2 2 22 2 2 2 2 2a b c ab bc ac 2 2 2a b c ab bc ac
![Page 59: 11X1 T03 01 inequations & inequalities (2011)](https://reader033.vdocuments.us/reader033/viewer/2022052904/55812bd3d8b42a68488b4a6d/html5/thumbnails/59.jpg)
(III) Squares are positive or zero2 2 2Show that if , and are positive, then a b c a b c ab bc ac
2 2 2a b ab
2 0a b 2 22 0a ab b
2 2 2a c ac 2 2 2b c bc
2 2 22 2 2 2 2 2a b c ab bc ac 2 2 2a b c ab bc ac
Exercise 3A; 4, 6ace, 7bdf, 8bdf, 9, 11, 12, 13ac, 15, 18bcd, 22, 24