1. Graphing and Solving Quadratic Inequalities
Algebra II Mr. Gilbert
Chapter 6.7
Graphing and Solving Quadratic Inequalities
Standard & Honors
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2. Agenda Warm up Homework Review Lesson New Homework Check your answers
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3. Click the mouse button or press the Space Bar to display the answers.
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Click the mouse button or press the Space Bar to display the answers.
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5. Homework Review
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6. 6.7 Graphing and solving Quadratic Inequalities
Example 1 Graph a Quadratic Inequality (4)
Example 2 Solve ax2 + bx + c  0 (3)
Example 3 Solve ax2 + bx + c  0 (5)
Example 4 Write an Inequality (4)
Example 5 Solve a Quadratic Inequality (5)
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Lesson 7 Contents

7. Graph
Step 1 Graph the related quadratic equation, Since the inequality symbol is >, the parabola should be dashed.
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Example 7-1a

8. Step 2 Test a point inside the parabola, such as (1, 2).
Graph
Step 2 Test a point inside the parabola, such as (1, 2).
(1, 2)
So, (1, 2) is a solution of the inequality.
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Example 7-1a

9. Step 3 Shade the region inside the parabola.
Graph
Step 3 Shade the region inside the parabola.
(1, 2)
(1, 2)
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Example 7-1a

10. Graph
Answer:
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Example 7-1b

11. Solve by graphing.
The solution consists of the x values for which the graph of the related quadratic function lies above the x-axis. Begin by finding the roots of the related equation.
Related equation
Factor.
Zero Product Property or
Solve each equation.
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Example 7-2a

12. Answer: The solution set is
Sketch the graph of the parabola that has x-intercepts at 3 and 1. The graph lies above the x-axis to the left of and to the right of
Answer: The solution set is
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Example 7-2a

13. Solve by graphing.
Answer:
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Example 7-2b

14. Use the Quadratic Formula.
Solve by graphing.
This inequality can be rewritten as The solution consists of the x-values for which the graph of the related quadratic equation lies on and above the x-axis. Begin by finding roots of the related equation.
Related equation
Use the Quadratic Formula.
Replace a with –2, b with –6 and c with 1.
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Example 7-3a

15. Simplify and write as two equations. or
Sketch the graph of the parabola that has x-intercepts of –3.16 and The graph should open down since a < 0.
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Example 7-3a

16. Answer:
The graph lies on and above the x-axis at and and between these two values. The solution set of the inequality is approximately
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Example 7-3a

17. Check Test one value of x less than –3. 16, one between –3. 16 and 0
Check Test one value of x less than –3.16, one between –3.16 and 0.16, and one greater than in the original inequality.
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Example 7-3a

18. Solve by by graphing.
Answer:
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Example 7-3b

19. Subtract 15 from each side.
Sports The height of a ball above the ground after it is thrown upwards at 40 feet per second can be modeled by the function where the height h(x) is given in feet and the time x is in seconds. At what time in its flight is the ball within 15 feet of the ground?
The function h(x) describes the height of the ball. Therefore, you want to find values of x for which
Original inequality
Subtract 15 from each side.
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Example 7-4a

20. Graph the related function using a graphing calculator.
The zeros are about 0.46 and The graph lies below the x-axis when or
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Example 7-4a

21. Answer: Thus, the ball is within 15 feet of the ground for the first 0
Answer: Thus, the ball is within 15 feet of the ground for the first 0.46 second of its flight and again after 2.04 seconds until the ball hits the ground at 2.5 seconds.
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Example 7-4a

22. Sports The height of a ball above the ground after it is thrown upwards at 28 feet per second can be modeled by the function where the height h(x) is given in feet and the time x is given in seconds. At what time in its flight is the ball within 10 feet of the ground?
Answer: The ball is within 10 feet of the ground for the first 0.5 second of its flight and again after 1.25 seconds until the ball hits the ground.
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Example 7-4b

23. First, solve the related equation .
Solve algebraically.
First, solve the related equation .
Related quadratic equation
Subtract 2 from each side.
Factor.
Zero Product Property or
Solve each equation.
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Example 7-5a

24. Plot –2 and 1 on a number line
Plot –2 and 1 on a number line. Use closed circles since these solutions are included. Notice that the number line is separated into 3 intervals.
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Example 7-5a

25. Test a value in each interval to see if it satisfies the original inequality.
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Example 7-5a

26. Answer: The solution set is This is shown on the number line below.
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Example 7-5a

27. Solve algebraically.
Answer:
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Example 7-5b

28. Homework - Honors See Syllabus 6.7
pp : 14, 24, 26, 28, even 43-49
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29. Homework See Syllabus 6.5 pp. 333-334: 15 – 39 multiples of 3
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