Inequalities & Interval Notation ES: Demonstrate understanding of concepts

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.

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.

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

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

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

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

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

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

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

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

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

Rational or Irrational ??? 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. 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

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

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

Inequalities Transitive Property

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

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

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

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

Inequalities Classic Mistake

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?

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

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

Interval Notation [ ]

) )

[ Always use parentheses with .

Interval Notation Always use parentheses with . )

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

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? Never use a bracket with 

Interval Notation What would the inequality notation look like?

Interval Notation What would the inequality notation look like?

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 - .

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