2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0)

Transcription

1 Quadratic Inequalities In One Variable LOOKS LIKE a quadratic equation but Doesn t have an equal sign (=) Has an inequality sign (>, <,, or ) SOLVE like quadratic equations GRAPH solutions on a # line FOR EXAMPLE: ax 2 + bx + c > 0 (a 0)

2 RECALL Inequalities AND statements BOTH conditions must be TRUE INTERSECTION Symbol

3 RECALL OR statements AT LEAST ONE condition must be TRUE UNION Symbol

4 RECALL Set-builder Notation {x x 5 or x > 5} Change or to USE interval notation

5 RECALL Set-builder Notation {x 5 x < 5} {x x > 5 and x < 5} Change and to USE interval notation

6 INTERVAL NOTATION < > BOUNDARY POINT IS NOT included ( ) BOUNDARY POINT IS included [ ]

7 For Example {x 5 x < 5} {x x 5 and x < 5} interval notation [ 5, 5)

8 For Example {x x 5 or x > 5} interval notation (, 5] (5, ) ALWAYS has ( )

9 Example 2A: Solving Quadratic Inequalities by Using Tables and Graphs Solve the inequality by using tables or graphs. x 2 + 8x Use a graphing calculator to graph each side of the inequality. Set Y 1 equal to x 2 + 8x + 20 and Y 2 equal to 5. Identify the values of x for which Y 1 Y 2.

10 Example 2A Continued The parabola is at or above the line when x is less than or equal to 5 or greater than or equal to 3. So, the solution set is x 5 or x 3 or (, 5] U [ 3, ). The table supports your answer. The number line shows the solution set

11 Example 2B: Solving Quadratics Inequalities by Using Tables and Graphs Solve the inequality by using tables and graph. x 2 + 8x + 20 < 5 Use a graphing calculator to graph each side of the inequality. Set Y 1 equal to x 2 + 8x + 20 and Y 2 equal to 5. Identify the values of which Y 1 < Y 2.

12 Example 2B Continued The parabola is below the line when x is greater than 5 and less than 3. So, the solution set is 5 < x < 3 or ( 5, 3). The table supports your answer. The number line shows the solution set

13 Check It Out! Example 2a Solve the inequality by using tables and graph. x 2 x + 5 < 7 Use a graphing calculator to graph each side of the inequality. Set Y 1 equal to x 2 x + 5 and Y 2 equal to 7. Identify the values of which Y 1 < Y 2.

14 Check It Out! Example 2a Continued The parabola is below the line when x is greater than 1 and less than 2. So, the solution set is 1 < x < 2 or ( 1, 2). The table supports your answer. The number line shows the solution set

15 Check It Out! Example 2b Solve the inequality by using tables and graph. 2x 2 5x Use a graphing calculator to graph each side of the inequality. Set Y 1 equal to 2x 2 5x + 1 and Y 2 equal to 1. Identify the values of which Y 1 Y 2.

16 Check It Out! Example 2b Continued The parabola is at or above the line when x is less than or equal to 0 or greater than or greater than or equal to 2.5. So, the solution set is (, 0] U [2.5, ) The number line shows the solution set

17 solve quadratic inequalities algebraically by finding the critical values X-intercepts

18 Example 3: Solving Quadratic Equations by Using Algebra Solve the inequality x 2 10x by using algebra. Step 1 Write the related equation. x 2 10x + 18 = 3

19 Example 3 Continued Step 2 Solve the equation for x to find the critical values. x 2 10x + 21 = 0 (x 3)(x 7) = 0 Factor. x 3 = 0 or x 7 = 0 x = 3 or x = 7 Write in standard form. Zero Product Property. Solve for x. The critical values are 3 and 7. The critical values divide the number line into three intervals: x 3, 3 x 7, x 7.

20 Example 3 Continued Step 3 Test an x-value in each interval. Critical values x 2 10x (2) 2 10(2) (4) 2 10(4) (8) 2 10(8) x x Test points Try x = 2. Try x = 4. Try x = 8.

21 Example 3 Continued Shade the solution regions on the number line. Use solid circles for the critical values because the inequality contains them. The solution is 3 x 7 or [3, 7]

22 Check It Out! Example 3a Solve the inequality by using algebra. x 2 6x Step 1 Write the related equation. x 2 6x + 10 = 2

23 Check It Out! Example 3a Continued Step 2 Solve the equation for x to find the critical values. x 2 6x + 8 = 0 (x 2)(x 4) = 0 Factor. x 2 = 0 or x 4 = 0 x = 2 or x = 4 Write in standard form. Zero Product Property. Solve for x. The critical values are 2 and 4. The critical values divide the number line into three intervals: x 2, 2 x 4, x 4.

24 Check It Out! Example 3a Continued Step 3 Test an x-value in each interval. Critical values x 2 6x (1) 2 6(1) (3) 2 6(3) (5) 2 6(5) x Try x = 1. Try x = 3. Try x = 5. Test points

25 Check It Out! Example 3a Continued Shade the solution regions on the number line. Use solid circles for the critical values because the inequality contains them. The solution is x 2 or x (, 2] U [4, )

26 Check It Out! Example 3b Solve the inequality by using algebra. 2x 2 + 3x + 7 < 2 Step 1 Write the related equation. 2x 2 + 3x + 7 = 2

27 Check It Out! Example 3b Continued Step 2 Solve the equation for x to find the critical values. 2x 2 + 3x + 5 = 0 ( 2x + 5)(x + 1) = 0 2x + 5 = 0 or x + 1 = 0 x = 2.5 or x = 1 Write in standard form. Factor. Zero Product Property. Solve for x. The critical values are 2.5 and 1. The critical values divide the number line into three intervals: x < 1, 1 < x < 2.5, x > 2.5.

28 Check It Out! Example 3b Continued Step 3 Test an x-value in each interval. Critical values 2x 2 + 3x + 7 < Test points 2( 2) 2 + 3( 2) + 7 < 2 2(1) 2 + 3(1) + 7 < 2 2(3) 2 + 3(3) + 7 < 2 x Try x = 2. Try x = 1. Try x = 3.

29 Check It Out! Example 3 Shade the solution regions on the number line. Use open circles for the critical values because the inequality does not contain or equal to. The solution is x < 1 or x > (, 1) U (2.5, )

30 HOMEWORK Textbook pages # s 5-9, 19 25, 29-33, ODD ONLY, all (matching)