Warm-up: Solve x2 – 3x + 2 = 0 HW: pg.73-75 (2, 6, 8, 10, 12, 28, 32, 42, 44, 50, 66, 70, 74, 76)
P.6 Solving Inequalities
Objective: Solve and graph linear inequalities. Solve and graph absolute value inequalities. Solve and graph polynomial inequalities. Write solutions in interval notation.
Greater than or equal to Inequality Symbols Less than Not equal to Less than or equal to Greater than or equal to Greater than
Ex: Solve the inequality. Flip the sign if multiplying or dividing by a negative number!
Graphing Linear Inequalities Remember: < and > signs will have ), ( or an open dot o and signs will have ], [ or a closed dot graph of graph of ) [ 4 5 6 7 -3 -2 -1 (-, 11/2) Unbounded [-2, ) Unbounded
Example: Solve and graph the solution. ] 6 7 8 9 (-, 7] Unbounded
Compound Inequality Example: Solve & graph. -18 ≤ 2t+8 ≤ 20 [ ] -13 6 [-13, 6] Bounded
Solve & graph. -6x + 9 < 3 or -3x -8 ≤ 13 -6x < -6 -3x ≤ 21 [ ( [ -7 1 [-7,) Unbounded
Absolute Value Inequalities Recall:
Absolute Value Inequalities Case 1 Example: Less th”and” and ( ) ( ) -2 8 (-2, 8) Bounded
Absolute Value Inequalities Case 2 Example: Great “or” or OR ] [ -5 4 (-, -5] U [4, ) Unbounded
Polynomial Inequalities
Definition of a Polynomial Inequality Polynomial Inequalities Definition of a Polynomial Inequality A polynomial inequality is any inequality that can be put in one of the forms where f is a polynomial.
Procedure for Solving Polynomial Inequalities 1) Express the inequality in the form f (x) < 0 or f (x) > 0 2) Solve the equation f (x) = 0. The real solutions are the boundary points. 3) Locate these boundary points on a number line, dividing the number line into intervals. 4) Choose one representative (test) number within each interval and check to see if it satisfies the inequality. 5) Write the solution set, selecting the interval(s) that satisfy the given inequality.
Polynomial Inequalities EXAMPLE Solve and graph the solution set on a number line: SOLUTION 1) Express the inequality in the form f (x) < 0. 2) Solve the equation f (x) = 0. The x-intercepts of f are 3 and 1. We will use these x-intercepts as boundary points on a number line.
Polynomial Inequalities 3) Locate the boundary points on a number line and separate the line into intervals. The number line with the boundary points is shown as follows: -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 The boundary points divide the number line into three intervals:
Polynomial Inequalities 4) Choose one representative (test) number within each interval and check to see if it satisfies the inequality. Interval Test Number Check Conclusion
Polynomial Inequalities CONTINUED 5) Write the solution set, selecting the interval(s) that satisfy the given inequality. Thus, the solution set of the given inequality, , is 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
Polynomial Inequalities EXAMPLE Solve and graph the solution set on a number line: SOLUTION 1) Express the inequality in the form f (x) 0.
Polynomial Inequalities 2) Solve the equation
Polynomial Inequalities 3) Locate the boundary points on a number line and separate the line into intervals. The number line with the boundary points is show as follows: -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 The boundary points divide the number line into three intervals:
Polynomial Inequalities 4) Choose one representative (test) number within each interval and check to see if it satisfies the inequality. Interval Test Number Check Conclusion
Polynomial Inequalities 5) Write the solution set Careful! Because the inequality involves , we must include the solutions of namely 2 and -2, in the solution set. Thus, the solution set of the given inequality, is 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
Summary: Solve and graph linear inequalities. Solve and graph absolute value inequalities. Solve and graph polynomial inequalities. Write solutions in interval notation.
Sneedlegrit: Solve x2 – 6 < x HW: pg.73-75 (2, 6, 8, 10, 12, 28, 32, 42, 44, 50, 66, 70, 74, 76)