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