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Inequalities & Interval Notation ES: Demonstrate understanding of concepts

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Objective To examine the properties of inequalities. To examine the properties of inequalities. To express inequalities in interval notation. To express inequalities in interval notation.

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Vocabulary Real Numbers: The set of numbers consisting of the positive numbers, the negative numbers, and zero. Real Numbers: The set of numbers consisting of the positive numbers, the negative numbers, and zero. Rational Number: A real number that can be expressed as a ratio of two integers. Rational Number: A real number that can be expressed as a ratio of two integers. Irrational Number: A real number that can not be expressed as a ratio of two integers. Irrational Number: A real number that can not be expressed as a ratio of two integers.

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Rational or Irrational ??? Rational Any rational number can be written as a fraction.

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Rational or Irrational ??? Rational Any integer can be written as a fraction.

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Rational or Irrational ??? Rational Any terminating decimal can be written as a fraction.

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Rational or Irrational ??? Rational Any repeating decimal can be written as a fraction.

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Rational or Irrational ??? Irrational Irrational numbers can be represented by decimal numbers in which the digits go on forever without ever repeating.

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Rational or Irrational ??? Irrational Some of the most common irrational numbers are radicals.

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Rational or Irrational ??? Rational Be careful, not all radicals are irrational.

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Rational or Irrational ??? Irrational Numbers containing are always irrational.

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Rational or Irrational ??? Rational Remember, any repeating decimal can be written as a fraction.

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Rational or Irrational ??? Irrational Numbers containing the mathematical constant e (Euler’s number 2.718) are always irrational.

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Vocabulary Real Number Line: A line that pictures real numbers as points. Real Number Line: A line that pictures real numbers as points. All real numbers (rational/irrational) can be graphed on a number line. All real numbers (rational/irrational) can be graphed on a number line. origin

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Inequalities Math Wild Kingdom The greedy crocodile always wants to eat the larger thing.

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Inequalities Less thanGreater than (smaller)(larger) Less thanGreater than (smaller)(larger) The arrow > points from the greater value to the lesser.

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Inequalities Transitive Property

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Inequalities What happens to the inequality sign when you add or subtract? The inequality remains the same.

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Inequalities Inequality sign is still correct What happens to the inequality sign when you multiply by 5?

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Inequalities Inequality sign is no longer correct What happens to the inequality sign when you multiply by -5?

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Inequalities Inequality sign must get flipped What happens to the inequality sign when you multiply by -5?

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Inequalities Classic Mistake

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Inequalities x exists between -3 and 2 x is larger than, but cannot equal -3 x is less than and can equal 2 < excludes the endpoint < includes the endpoint What does this mean? What x-values is it talking about?

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Inequalities x exists between -3 and 2 ( ] Parentheses: endpoint is not allowed as a value Bracket: endpoint is allowed as a value

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Interval Notation x exists between -3 and 2 ( ] Same as Interval excludes -3, and includes 2

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Interval Notation [ ]

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) )

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[ Always use parentheses with .

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Interval Notation Always use parentheses with . )

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Interval Notation What values below does this expression represent? All values Nothing 0 (-1, 1)

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Interval Notation What values does this expression represent? This represents all values on the line.

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Interval Notation What does this mean? Is there anything wrong with the notation? Never use a bracket with

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Interval Notation What would the inequality notation look like?

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Interval Notation What would the inequality notation look like?

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Conclusion When increasing/decreasing two sides of an inequality by the same amount, the inequality remains. When increasing/decreasing two sides of an inequality by the same amount, the inequality remains. When multiplying/dividing an inequality by a negative, the inequality sign flips. When multiplying/dividing an inequality by a negative, the inequality sign flips. Use a bracket if the inequality symbol next to the number is, otherwise use a parenthesis. Use a bracket if the inequality symbol next to the number is, otherwise use a parenthesis. Always use parentheses with and - . Always use parentheses with and - .

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Exit Slip: Answer the below questions on the note card then turn in. Make sure your name is on it. 1) Circle all that apply: a) -5 is… Real Rational Irrational b) is… Real Rational Irrational c) is… Real Rational Irrational 2) Write the interval notation for each of the below a) - 4 < xb) 2 < x ≤ 5c) x ≥ 0 3) Write the interval notation for the graph below which represents all real numbers 4) Solve the inequality and write the solution in interval notation -3x + 2 < 11 0

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