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Inequalities & Interval Notation ES: Demonstrate understanding of concepts Objective To examine the properties of inequalities. To express inequalities in interval notation. Vocabulary 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. Irrational Number: A real number that can not be expressed as a ratio of two integers. Rational or Irrational ??? 3 4 Rational Any rational number can be written as a fraction. Rational or Irrational ??? 365 365 1 Rational Any integer can be written as a fraction. Rational or Irrational ??? 5 0.625 8 Rational Any terminating decimal can be written as a fraction. Rational or Irrational ??? 2 0.222 9 Rational Any repeating decimal can be written as a fraction. Rational or Irrational ??? 1.732050808... Irrational Irrational numbers can be represented by decimal numbers in which the digits go on forever without ever repeating. Rational or Irrational ??? 5 Irrational Some of the most common irrational numbers are radicals. Rational or Irrational ??? 3 125 5 Rational Be careful, not all radicals are irrational. Rational or Irrational ??? 2 3 Irrational Numbers containing are always irrational. Rational or Irrational ??? 1 0.142857 7 Rational Remember, any repeating decimal can be written as a fraction. Rational or Irrational ??? 8e Irrational Numbers containing the mathematical constant e (Euler’s number 2.718) are always irrational. Vocabulary Real Number Line: A line that pictures real numbers as points. All real numbers (rational/irrational) can be graphed on a number line. 32 2.6 5 4 3 2 origin 1 0 e 2 1 2 3 4.3 4 5 Inequalities Math Wild Kingdom The greedy crocodile always wants to eat the larger thing. Inequalities A B Less than Greater than (smaller) (larger) A B Greater than Less than (larger) (smaller) The arrow > points from the greater value to the lesser. Inequalities Transitive Property A B B C A C Inequalities A B AC BC What happens to the inequality sign when you add or subtract? The inequality remains the same. AC B C Inequalities 3 2 What happens to the inequality sign when you multiply by 5? 3 5 2 5 15 10 Inequality sign is still correct Inequalities 3 2 What happens to the inequality sign when you multiply by -5? 3 5 2 5 15 10 Inequality sign is no longer correct Inequalities 3 2 What happens to the inequality sign when you multiply by -5? 3 5 2 5 15 10 Inequality sign must get flipped Inequalities Classic Mistake Inequalities What does this mean? What x-values is it talking about? 3 x 2 x exists between -3 and 2 x is less than and can equal 2 x is larger than, but cannot equal -3 < excludes the endpoint < includes the endpoint Inequalities 3 x 2 ( 3 x exists between -3 and 2 ] 2 Parentheses: endpoint is not allowed as a value Bracket: endpoint is allowed as a value Interval Notation 3 x 2 x exists between -3 and 2 ] ( 3 Same as 2 ( 3 , 2 ] Interval excludes -3, and includes 2 Interval Notation 1 x 10 [ ] 1 10 [ 1 , 10 ] Interval Notation 3 x 0 ) ) 3 0 ( 3 , 0 ) Interval Notation x5 [ 5 [ 5 , ) Always use parentheses with . Interval Notation x 1 ) 1 ( , 1) Always use parentheses with . Interval Notation What values below does this expression represent? , 0 (-1, 1) Nothing All values Interval Notation What values does this expression represent? , This represents all values on the line. Interval Notation What does this mean? Is there anything wrong with the notation? , 4 Never use a bracket with Interval Notation 5, 2 What would the inequality notation look like? 5 x 2 Interval Notation 0, 1 What would the inequality notation look like? 0 x 1 Conclusion 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. Use a bracket if the inequality symbol next to the number is < or >, otherwise use a parenthesis. Always use parentheses with and - . 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 b) c) 3 is… is… 81 Real Real Rational Rational Irrational Irrational Irrational 2) Write the interval notation for each of the below a) - 4 < x b) 2 < x ≤ 5 c) x ≥ 0 3) Write the interval notation for the graph below which represents all real numbers 0 4) Solve the inequality and write the solution in interval notation -3x + 2 < 11