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Solving Inequalities

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Solving Inequalities Vocabulary 1) set-builder notation 2) interval notation <ul><li>Solve inequalities. </li></ul>

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For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities

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For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities

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For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities

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For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities

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For any two real numbers, a and b , exactly one of the following statements is true . This is known as the Trichotomy Property Solving Inequalities

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For any two real numbers, a and b , exactly one of the following statements is true . This is known as the Trichotomy Property or the property of order . Solving Inequalities

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Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Solving Inequalities

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Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities

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Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b

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Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c c

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Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c c

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Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c

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Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c

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Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c

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Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities

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Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b

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Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b c

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Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b c

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Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b

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Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b

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Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b

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Solving Inequalities a

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Solving Inequalities a

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Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a

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Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a

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Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a

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Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. We use a closed circle (dot) to indicate that a IS part of the solution set. a a

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Solving Inequalities a

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Solving Inequalities a

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Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a

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Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a

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Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a

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Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. We use a closed circle (dot) to indicate that a IS part of the solution set. a a

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Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Solving Inequalities

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Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities

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Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities

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Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive:

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Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive:

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Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive:

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Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive: c is negative:

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Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive: c is negative:

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Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive: c is negative:

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Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? Solving Inequalities

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Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? Solving Inequalities HINT: is the same as

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Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? Solving Inequalities HINT: is the same as

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Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? So, see rules for multiplication! Solving Inequalities HINT: is the same as

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The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities 4

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The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities 4 set-builder notation

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The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is less than 4 } 4 set-builder notation

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The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is less than 4 } 4 set-builder notation Identify the variable used

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The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is less than 4 } 4 set-builder notation Identify the variable used Describe the limitations or boundary of the variable

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The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities -7

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The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities -7 set-builder notation

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The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is greater than or equal to negative 7 } -7 set-builder notation

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The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is greater than or equal to negative 7 } Identify the variable used -7 set-builder notation

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The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is greater than or equal to negative 7 } Identify the variable used Describe the limitations or boundary of the variable -7 set-builder notation

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The solution set of an inequality can also be described by using interval notation . Solving Inequalities

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The solution set of an inequality can also be described by using interval notation . Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively.

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The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively.

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The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4

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The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation

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The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation

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The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation -7

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The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation -7 interval notation

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End of Lesson Solving Inequalities

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Credits PowerPoint created by Using Glencoe’s Algebra 2 text, © 2005 Robert Fant http://robertfant.com