Inequations & Inequalities
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
Inequations & Inequalities2e.g. ( ) 5 6 0i x x
1. Quadratic Inequations
6 1 0x x
y
x1–6
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?
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?
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
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
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
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
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?
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?
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 > ?
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
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 > ?
2. Inequalities with Pronumerals in the Denominator
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2. Inequalities with Pronumerals in the Denominator1e.g. ( ) 3ix 1) Find the value where the denominator is zero 0x
2) Solve the “equality”
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
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
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
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
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
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
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
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
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. 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
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. 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. 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
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
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
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
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
3. Proving Inequalities
3. Proving Inequalities(I) Start with a known result
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
3. Proving Inequalities(I) Start with a known result
If prove 2
x zx y y z y
x y y z
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
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
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
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
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
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
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
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
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
(III) Squares are positive or zero
(III) Squares are positive or zero2 2 2Show that if , and are positive, then a b c a b c ab bc ac
(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
(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
(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
(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
(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
(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
(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
