special equations : and / or and quadratic inequalities and / or are logic operators. and – where...
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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