Warm-up: Solve the inequality and graph the solution set. x3 + 2x2 – 9x  18 HW: pg.73-75 (4, 5, 7, 9, 11, 30, 34, 46, 52, 68, 80, 81, 82, 84, 86, 88)

 

Objective: Check Polynomial Equation solutions with the TI83/84 calculator Solve and graph rational inequalities. Write solutions in interval notation.

Review Polynomial Inequalities Solve: Put in standard form: Replace the > with an = and solve to get boundaries

The solution is x > 2 written in interval notation is Put the solutions (x = -2 or x = 2) to the equation on a number line to get intervals. (-, -2) , (-2, 2) , (2, ) Pick a test point in each interval formed and determine if the interval is in the solution. The solution is x > 2 written in interval notation is

TI83/84 Calculator We can get to the same conclusion using the TI83/84 calculator. Method: Put the polynomial equation in standard form Enter the left side as a function [y=] Find the zeros Examine the graph to determine position or negative f(x) values

Find the zeroes: x = -2 or x = 2 Examine the graph to see that the positive values occur when x > 2 This is verified in the Table.

Solve Solution: x3 – 6x2 + 3x + 10 ≤ 0 The solution set is

Rational Inequalities EXAMPLE Solve and graph the solution set: SOLUTION 1) Express the inequality so that one side is zero and the other side is a single quotient and simplify! This inequality is equivalent to the one we wish to solve. It is in the form f (x) < 0, where

Rational Inequalities 2) Set the numerator and the denominator of f equal to zero. The real solutions are the boundary points. We will use these solutions as boundary points on a number line. 3) Locate the boundary points on a number line and separate the line into intervals. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 The boundary points divide the number line into three intervals:

Rational Inequalities 4) Choose one representative (test) number within each interval and evaluate f at that number. Interval Test Number Check Conclusion

Rational Inequalities CONTINUED 5) Write the solution set, selecting the interval(s) that satisfy the given inequality. The graph of the solution set on a number line is shown as follows: ) ( -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Example: Solve the inequality -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Pick a test point for each interval

Since this inequality involves equality, we include – 7. Since (-, 7) and (4, ) are part of the solution we shade the regions. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Since this inequality involves equality, we include – 7. The point, x = 4, cannot be included since it is a domain restriction! The solution set is

Summary: Check Polynomial Equation solutions with the TI83/84 calculator Solve and graph rational inequalities. Write solutions in interval notation.

  HW: pg.73-75 (4, 5, 7, 9, 11, 30, 34, 46, 52, 68, 80, 81, 82, 84, 86, 88)