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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

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

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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 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.

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Special Equations : AND / OR and Quadratic Inequalities EXAMPLE :

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Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : 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

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Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : 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

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Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : 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

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Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : 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

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

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

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

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

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

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Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : 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 This shared space is our final graph

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

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

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

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Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3: 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 Graph the solution set for each

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

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

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

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

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

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

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

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

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

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

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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

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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

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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

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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 - 23 Open Circle

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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 - 23 Open Circle TTTTTTTTTTTTTTTFFFFFF

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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 - 23 Open Circle TTTTTTTTTTTTTTTFFFFFF ALWAYS graph the TRUE sections… Answer as an interval

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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

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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

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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

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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 FFFFF TTTTTTTTTTTTTTTTTTTTTT ALWAYS graph the TRUE sections…

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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 FFFFF TTTTTTTTTTTTTTTTTTTTTT ALWAYS graph the TRUE sections… Answer as an interval

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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 0

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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 0

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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

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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 FTFT

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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 FTFT ALWAYS graph the TRUE sections…

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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 FTFT ALWAYS graph the TRUE sections… Answer as an interval

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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

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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

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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 35 Open Circle

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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 35 Open Circle F TT F

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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 35 Open Circle F TT F ALWAYS graph the TRUE sections…

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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 35 Open Circle F TT F ALWAYS graph the TRUE sections… Answer as an interval

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