Special Equations : AND / OR and Quadratic Inequalities AND / OR are logic operators. AND – where...

Special Equations : AND / OR and Quadratic Inequalities

AND / OR are logic operators.

AND – where two solution sets “share” common elements.

- similar to intersection of two sets

OR – where two solution sets are merged together

- similar to union of two sets

Special Equations : AND / OR and Quadratic Inequalities

AND / OR are logic operators.

AND – where two solution sets “share” common elements.

- similar to intersection of two sets

OR – where two solution sets are merged together

- similar to union of two sets

When utilizing these in graphing multiple inequality equations, a number line graph helps to “see” the final solution.

Special Equations : AND / OR and Quadratic Inequalities

AND / OR are logic operators.

AND – where two solution sets “share” common elements.

- similar to intersection of two sets

OR – where two solution sets are merged together

- similar to union of two sets

When utilizing these in graphing multiple inequality equations, a number line graph helps to “see” the final solution.

We will first look at how they are different.

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE : 5 AND 3for set solution theShow xx

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE : 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

- 3 5

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE : 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution set for each

- 3 5

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE : 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution set for each

3. Find where they “share” elements

- where are they “on top” of each other

- in this case they share between (– 3) and 5

- 3 5

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE : 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution set for each

3. Find where they “share” elements

- in this case they share between (– 3) and 5

- where are they “on top” of each other

4. This shared space is our final graph

- 3 5

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE : 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution set for each

3. Find where they “share” elements

- in this case they share between (– 3) and 5

- where are they “on top” of each other

4. This shared space is our final graph

- 3 5

Answer as an interval

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 2 : 5 AND 3for set solution theShow xx

Special Equations : AND / OR and Quadratic Inequalities

5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

- 3 5

EXAMPLE # 2 :

Special Equations : AND / OR and Quadratic Inequalities

5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution set for each

- 3 5

EXAMPLE # 2 :

Special Equations : AND / OR and Quadratic Inequalities

5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution set for each 3. Find where they “share” elements - where are they “on top” of each other - in this case they share numbers greater than 5

- 3 5

EXAMPLE # 2 :

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE : 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution set for each 3. Find where they “share” elements - where are they “on top” of each other - in this case they share numbers greater than 5

- 3 5

4. This shared space is our final graph

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE : 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution set for each 3. Find where they “share” elements - where are they “on top” of each other - in this case they share numbers greater than 5

- 3 5

4. This shared space is our final graph

Answer as an interval

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 3 : 5 AND 3for set solution theShow xx

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 3 : 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

- 3 5

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 3: 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

- 3 5

2. Graph the solution set for each

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 3 : 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

- 3 5

2. Graph the solution set for each3. Find where they “share” elements - where are they “on top” of each other - in this case they DO NOT share elements

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 3 : 5 AND 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

- 3 5

2. Graph the solution set for each

3. Find where they “share” elements - where are they “on top” of each other - in this case they DO NOT share elements

4. SO we have An EMPTY SET

Ø

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 4 : 5 OR 3for set solution theShow xx

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 4 : 5 OR 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

- 3 5

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 4 : 5 OR 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution for each

- 3 5

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 4 : 5 OR 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution for each

3. Now merge the two graphs and keep everything

- this will be your answer

- 3 5

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 4 : 5 OR 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution for each

3. Now merge the two graphs and keep everything

- this will be your answer

- 3 5

Answer as an interval

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 5 : 5 OR 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution for each

- 3 5

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 5 : 5 OR 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution for each

3. Now merge the two graphs and keep everything

- this will be your answer

- 3 5

Special Equations : AND / OR and Quadratic Inequalities

EXAMPLE # 5 : 5 OR 3for set solution theShow xx

STEPS : 1. Create a number line and locate your points.

( open circle for 5 and closed for – 3 )

- when graphing, graph one point higher than the other

2. Graph the solution for each

3. Now merge the two graphs and keep everything

- this will be your answer

- 3 5

Answer as an interval

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Example # 1 : Graph the solution set for 062 xx

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Example # 1 : Graph the solution set for 062 xx

2 3

02 03 023

xx

xxxx

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Example # 1 : Graph the solution set for 062 xx

2 3

02 03 023

xx

xxxx

- 2 3

Open Circle

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Example # 1 : Graph the solution set for 062 xx

2 3

02 03 023

xx

xxxx

- 2 3

Open Circle

TRUE 66

6600

0 : TEST2

x

TTTTTTTTTTTTTTT FFFFFFFFFFFF

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Example # 1 : Graph the solution set for 062 xx

2 3

02 03 023

xx

xxxx

- 2 3

Open Circle

TRUE 66

6600

0 : TEST2

x

TTTTTTTTTTTTTTT FFFFFFFFFFFFALWAYS graph the TRUE sections…

Answer as an interval

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Example # 1 : Graph the solution set for 01272 xx

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Example # 1 : Graph the solution set for 01272 xx

4,3 043 xxxx

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Example # 1 : Graph the solution set for 01272 xx

4,3 043 xxxx

- 4 - 3

Closed Circle

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Example # 1 : Graph the solution set for 01272 xx

4,3 043 xxxx

- 4 - 3

Closed Circle

TRUE 012

012070

0Test 2

x

FFFFF TTTTTTTTTTTTTTTTTTTTTT

ALWAYS graph the TRUE sections…

Special Equations : AND / OR and Quadratic Inequalities

Graphing Quadratic Inequalities :

1. Factor to find “critical points” ( where the equation = 0 )

2. Locate your points on a number line

3. Pick a test point for TRUE or FALSE

- true / false changes every time you pass a critical point

Example # 1 : Graph the solution set for 01272 xx

4,3 043 xxxx

- 4 - 3

Closed Circle

TRUE 012

012070

0Test 2

x

FFFFF TTTTTTTTTTTTTTTTTTTTTT

ALWAYS graph the TRUE sections…

Answer as an interval

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Example # 1 : Graph the solution set for 318

x

x

0

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Next, multiply EVERYTHING by “x” and get all terms on one side…

Example # 1 : Graph the solution set for

0183

318

318

2

2

xx

xx

xxx

xx

0

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Next, multiply EVERYTHING by “x” and get all terms on one side…

Now just follow the steps as we did before…

Example # 1 : Graph the solution set for

0183

318

318

2

2

xx

xx

xxx

xx

0

3,6

036

xx

xx

-3 6

Closed Circle

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Next, multiply EVERYTHING by “x” and get all terms on one side…

Now just follow the steps as we did before…

** I like to test 1000 in these…because 18/1000 is very small, you can disregard it !!!

Example # 1 : Graph the solution set for

0183

318

318

2

2

xx

xx

xxx

xx

0

3,6

036

xx

xx

-3 6

Closed Circle

1Test x

FALSE 31000

31000

181000

1000Test

x

FTFT

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Next, multiply EVERYTHING by “x” and get all terms on one side…

Now just follow the steps as we did before…

** I like to test 1000 in these…because 18/1000 is very small, you can disregard it !!!

Example # 1 : Graph the solution set for

0183

318

318

2

2

xx

xx

xxx

xx

0

3,6

036

xx

xx

-3 6

Closed Circle

1Test x

FALSE 31000

31000

181000

1000Test

x

FTFTALWAYS graph the TRUE sections…

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Next, multiply EVERYTHING by “x” and get all terms on one side…

Now just follow the steps as we did before…

** I like to test 1000 in these…because 18/1000 is very small, you can disregard it !!!

Example # 1 : Graph the solution set for

0183

318

318

2

2

xx

xx

xxx

xx

0

3,6

036

xx

xx

-3 6

Closed Circle

1Test x

FALSE 31000

31000

181000

1000Test

x

FTFTALWAYS graph the TRUE sections…

Answer as an interval

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Example # 2 : Graph the solution set for

0

815

x

x

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Next, multiply EVERYTHING by “x” and get all terms on one side…

Example # 2 : Graph the solution set for

0

0158

815

815

2

2

xx

xx

xxx

xx

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Next, multiply EVERYTHING by “x” and get all terms on one side…

Factor and get critical points…

Example # 2 : Graph the solution set for

0

0158

815

815

2

2

xx

xx

xxx

xx

3,5

035

xx

xx

3 5

Open Circle

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Next, multiply EVERYTHING by “x” and get all terms on one side…

Factor and get critical points…

Test 1000…

Example # 2 : Graph the solution set for

0

0158

815

815

2

2

xx

xx

xxx

xx

3,5

035

xx

xx

3 5

Open Circle

FALSE 81000

81000

151000

FT TF

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Next, multiply EVERYTHING by “x” and get all terms on one side…

Factor and get critical points…

Test 1000…

Example # 2 : Graph the solution set for

0

0158

815

815

2

2

xx

xx

xxx

xx

3,5

035

xx

xx

3 5

Open Circle

FALSE 81000

81000

151000

FT TFALWAYS graph the TRUE sections…

Special Equations : AND / OR and Quadratic Inequalities

In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it…

Next, multiply EVERYTHING by “x” and get all terms on one side…

Factor and get critical points…

Test 1000…

Example # 2 : Graph the solution set for

0

0158

815

815

2

2

xx

xx

xxx

xx

3,5

035

xx

xx

3 5

Open Circle

FALSE 81000

81000

151000

FT TFALWAYS graph the TRUE sections…

Answer as an interval