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 :

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : - 3 5   Answer as an interval 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 2 :

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 2 : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 2 : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 2 : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : - 3 5 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 4. This shared space is our final graph

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : - 3 5   Answer as an interval 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 4. This shared space is our final graph

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3 :

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3 : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3: - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3 : - 3 5 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 DO NOT share elements

Ø Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3 : Ø - 3 5 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 DO NOT share elements 4. SO we have An EMPTY SET

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 4 :

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 4 : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 4 : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 4 : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 4 : - 3 5   Answer as an interval 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 5 : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 5 : - 3 5 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

Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 5 : - 3 5   Answer as an interval 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

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

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

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 Open Circle - 2 3

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 Open Circle FFFFFF TTTTTTTTTTTTTTT FFFFFF - 2 3

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 Open Circle   Answer as an interval ALWAYS graph the TRUE sections… FFFFFF TTTTTTTTTTTTTTT FFFFFF - 2 3

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

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

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 Closed Circle - 4 - 3

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 Closed Circle ALWAYS graph the TRUE sections… TTTTT FFFFF TTTTTTTTTTTTTTTTT - 4 - 3

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 Closed Circle   Answer as an interval ALWAYS graph the TRUE sections… TTTTT FFFFF TTTTTTTTTTTTTTTTT - 4 - 3

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

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

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 Closed Circle -3 6

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 Closed Circle T F T F -3 6

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 Closed Circle ALWAYS graph the TRUE sections… T F T F -3 6

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 Closed Circle   Answer as an interval ALWAYS graph the TRUE sections… T F T F -3 6

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

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

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 Open Circle 3 5

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 Open Circle T F T F 3 5

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 Open Circle ALWAYS graph the TRUE sections… T F T F 3 5

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 Open Circle   Answer as an interval ALWAYS graph the TRUE sections… T F T F 3 5