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WARM-UP : Express the following in interval notation: a) b) c)

L5 – Solving Inequalities. WARM-UP : Express the following in interval notation: a) b) c) 2. Solve the following equation: . ? Question ?. The path of a toy rocket can be modelled by the equation

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WARM-UP : Express the following in interval notation: a) b) c)

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  1. L5 – Solving Inequalities WARM-UP: Express the following in interval notation: a) b) c) 2. Solve the following equation: . .

  2. ? Question ? The path of a toy rocket can be modelled by the equation where h, is the height of the rocket in metres, and t, is the time elapsed after the rocket has been launched, in seconds. Over what time interval is the height above 3m? The height of the rocket is above 3m when

  3. Polynomial Inequalities • Often, instead of comparing two polynomial functions for equality I.e. f (x) = g(x) • we need to compare the two polynomials in terms of magnitude. I.e. When is f (x) > g(x), or f (x) < g(x) • As a result, the solution to a polynomial inequality could be a value of x on a given interval that satisfies the inequality. • These intervals would be the solutions to the polynomial inequality.

  4. Polynomial Inequalities • A polynomial inequality results when the equal sign in a polynomial equation is replaced with an inequality symbol. • Inequality Symbols: ; ; ; • Polynomial inequalities may be solved graphically by determining: • The x-intercepts of the “equation” and then using the graph to determine the intervals that satisfy the inequality. • The points of intersection of the functions on either side of the inequality

  5. Polynomial Inequalities EXAMPLE 1: Consider the function • State the domain and range of . • Find the real zeroes of . • Graph the function. • Find the interval(s) where .

  6. Polynomial Inequalities EXAMPLE 2:Given the graph below, write • The x-intercepts • The intervals of xfor which the graph is positive. • The intervals of xfor which the graphis negative.

  7. Polynomial Inequalities EXAMPLE 3:Solve the following polynomial inequality graphically:

  8. Interval Charts • When solving polynomial inequalities algebraically, an interval chartis a quick and efficient method to use. • An interval chart allows you to determine: • The end behaviours of the polynomial • The behaviour of the polynomial between the roots • To use an interval chart: • Write the polynomial in factored form • Choose a value of xwithin each interval • Substitute that value of x in each factor to determine if the result will be a positive or negative value • Determine the behaviour of the polynomial using the result of each factor within the interval

  9. Polynomial Inequalities EXAMPLE 3:Solve the following polynomial inequality algebraically : Set the inequality “equal” to zero Factor Roots: Let:

  10. Polynomial Inequalities EXAMPLE 3: (Continued) - - - -  - - + + + - + -  + + + +

  11. Polynomial Inequalities EXAMPLE 4:Solve the following polynomial inequality graphically. Then verify your answer by solving the polynomial inequality algebraically.

  12. HomeFUN!!!  • Advanced Functions 12 Textbook! • Pages 129 – 131 # 1 – 3, 5, 6 (Don’t do #10) REMINDER • Portfolio Assignment: Due Wed Oct 3rd • Chapter 2 KAC Test: Thurs Oct 4th • Unit 1 TC Assessment: Fri Oct 5th

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