McGraw-Hill Ryerson

1. Click here to begin the lesson McGraw-Hill Ryerson Pre-Calculus 11 Chapter 9 Linear and Quadratic Inequalities

2. McGraw-Hill Ryerson Pre-Calculus 11 Teacher Notes 1. This lesson is designed to help students conceptualize the main ideas of Chapter 9. 2. To view the lesson, go to Slide Show > View Show (PowerPoint 2003). 3. To use the pen tool, view Slide Show, click on the icon in the lower left of your screen and select Ball Point Pen. 4. To reveal an answer, click on or follow the instructions on the slide. To reveal a hint, click on . To view the complete solution, click on the View Solution button. Navigate through the lesson using the and buttons. 5. When you exit this lesson, do not save changes.

3. Linear Inequalities Chapter 9 The graph of the linear equation x – y = –2 is referred to as a boundary line. This line divides the Cartesian plane into two regions: For one region, the condition x – y < –2 is true. For the other region, the condition x – y > –2 is true. Use the pen to label the conditions below to the corresponding parts of the graph on the Cartesian plane. x–y = –2 x–y < –2 x–y > –2 x–y = –2 x–y < –2 x–y > –2

4. Linear Inequalities Chapter 9 The ordered pair (x, y) is a solution to a linear inequality if its coordinates satisfy the condition expressed by the inequality. Which of the following ordered pairs (x, y) are solutions of the linear inequality x – 4y < 4? Click on the ordered pairs to check your answer. Use the pen tool to graph the boundary line and plot the points on the graph. Then, shade the region that represents the inequality. Click here for the solution.

5. Graphing Linear Inequalities Chapter 9 Match the inequality to the appropriate graph of a boundary line below. Complete the graph of each inequality by shading the correct solution region. Match Shade

6. Graphing a Linear Inequality Chapter 9 Use the pen tool to graph the following inequalities. Describe the steps required to graph the inequality. a) Click here for the solution.

7. Graphing a Linear Inequality Chapter 9 Match each inequality to its graph. Then, click on the graph to check the answer.

8. Linear Inequalities Chapter 9 Write an inequality that represents each graph. 2. 1. (2, 4) (0, 3) 0 0 (0, -2) (2, -1)

9. Solve an Inequality Chapter 9 Paul is hosting a barbecue and has decided to budget $48 to purchase meat. Hamburger costs $5 per kilogram and chicken costs $6.50 per kilogram. Write an inequality to represent the number of kilograms of each that Paul may purchase. Let h = kg of hamburger c = kg of chicken Write the equation of the boundary line below and draw its graph. Chicken Shade the solution region for the inequality. Hamburger Click here for the solution.

10. Solve an Inequality Chapter 9 c Chicken Hamburger 1. Can Paul buy 6 kg of hamburger and 4 kg chicken if he wants to stay within his set budget? h No 2. How many kilograms of chicken can Paul buy if he decides not to buy any hamburger? 7.38 kg 3. If Paul buys 3 kg of hamburger, what is the greatest number of kilograms of chicken he can buy? 5.08 kg Click here for the solution.

11. Quadratic Inequalities Chapter 9 Solve x2–x– 12 > 0. Use the pen tool. Solve the related equation to determine intervals of numbers that may be solutions of the inequality. Plot the solutions on a number line creating the intervals for investigation. Pick a number from each interval to test in the original inequality. If the number tested satisfies the inequality, then all of the numbers in that interval are solutions. -5 -4 -3 -2 -1 1 2 5 0 3 4 State the solution set. -5 -4 -3 -2 -1 1 2 5 0 3 4 Click here for the solution.

12. Quadratic Inequalities Chapter 9 Solve x2–x– 12 > 0. Use the pen tool. Graph the corresponding quadratic function y = x2–x– 12 to verify your solution from the previous page. Click here for the solution.

13. Chapter Sign Analysis 9 Solve x2– 3x– 4 > 0. Use the pen tool to solve the related quadratic equation to obtain the boundary points for the intervals. Use the boundary points to mark off test intervals on the number line. x2 - 3x - 4 = 0 1. 1. Determine the intervals when each of the factors is positive or negative. x - 4 2. x - 4 x + 1 x + 1 Determine the solution using the number line. (x - 4)(x + 1) (x - 4)(x + 1) 3. 4. Click here for the solution.

14. Chapter Sign Analysis 9 Solve x2– 3x– 4 > 0. Use the pen tool to create a graph of the related function to confirm your solutions. x2 - 3x - 4 = 0 1. 1. x - 4 2. x + 1 (x - 4)(x + 1) Click here for the solution. 3. 4.

15. Graphing a Quadratic Inequality Chapter 9 Choose the correct shaded region to complete the graph of the inequality. Circle your choice using the pen tool.

16. Quadratic Inequality in Two Variables Chapter 9 Match each inequality to its graph using the pen tool.

17. Click here to return to the start The following pages contain solutions for the previous questions.

18. Solutions (0, 4) 0 (4, 0) (-4, 0) (0, 0) (0, -4) Go back to the question.

19. Solutions An example method for graphing an inequality would be: • The x-intercept is the point (–2, 0), the y-intercept is the point (0, –4). • The inequality is greater than and equal to. Therefore, the boundary line is a solid line. • Use a test point (0, 0). The point makes the inequality true. • Therefore, shade above the line. • Slope of the line is . and the y-intercept is the point (0, 1). • The inequality is less than. Therefore, the boundary line is a broken line. • Use a test point (0, 0). The point makes the inequality true. • Therefore, shade below the line. Go back to the question.

20. Solutions Let h = kg of hamburger c = kg of chicken Write an inequality to represent the number of kilograms of each that Paul may purchase. Graph the boundary line for the inequality. Chicken c h Hamburger Go back to the question.

21. Solutions c (0, 7.38) (3, 5) Chicken (6, 4) 1. Can Paul buy 6 kg of hamburger and 4 kg chicken if he wants to stay within his set budget? h 2. How many kilograms of chicken can Paul buy if he decides not to buy any hamburger? Hamburger 3. If Paul buys 3 kg of hamburger, what is the greatest whole number of kilograms of chicken he can buy? The point (6, 4) is not within the shaded region. Paul could not purchase 6 kg of hamburger and 4 kg of chicken. This would be the point (3, 5). Paul could buy 5 kg of chicken. This is the point (0, 7.38). Buying no hamburger would be the y-intercept of the graph. Go back to the question.

22. Solutions Solve the related equation to determine intervals of numbers that may be solutions of the inequality. Plot the solutions on a number line creating the intervals for investigation. Pick a number from each interval to test in the original inequality. If the number tested satisfies the inequality, then all of the numbers in that interval are solutions. -5 -4 -3 -2 -1 1 2 5 0 3 4 Test point 5: (5)2 - (5) - 12 > 0 True Test point 0: (0)2 - (0) - 12 > 0 False Test point -5: (-5)2 - (-5) - 12 > 0 True The solution set is {x | x < –3 or x > 4, x R}. State the solution set. -5 -4 -3 -2 -1 1 2 5 0 3 4 Go back to the question.

23. Solutions Solve x2–x– 12 > 0 The inequality may have been solved by examining the graph of the corresponding function, y = x2 – x – 12. The quadratic inequality is greater than zero where the graph is above the x-axis. x < –3 or x > 4 Go back to the question.

24. Solutions Solve the related quadratic equation to obtain the boundary points for the intervals. x2– 3x– 4 = 0 (x– 4)(x + 1) = 0 x – 4 = 0 or x + 1 = 0 x = 4 x = –1 x2 - 3x - 4 = 0 1. 1. Use the boundary points to mark off test intervals on the number line. Determine the intervals when each of the factors is positive or negative. – – + x - 4 2. x– 4 – x + 1 + + x + 1 – + + Determine the solution using the number line. (x - 4)(x + 1) (x – 4)(x + 1) 3. 4. x < –1 or x > 4 Go back to the question.

25. Solutions A graph of the related function may be used to confirm your solutions. x2 - 3x - 4 = 0 1. 1. x - 4 2. x < –1 or x > 4 x + 1 Go back to the question. (x - 4)(x + 1) 3. 4.