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McGraw-Hill Ryerson
Pre-Calculus 11
Chapter 9
Linear and Quadratic Inequalities
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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. Chapter
Linear Inequalities 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. Click here for the solution.
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.
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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)
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7. Graphing a Linear Inequality
Chapter 9
Match each inequality to its graph.
Then, click on the graph to check the answer.

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

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Chapter
Solve an Inequality 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
Chicken
Hamburger
Write the equation of the boundary line below and draw its graph.
Shade the solution region for the inequality.
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10. Click here for the solution.
Solve an Inequality
Chapter 9
Hamburger
Chicken h c
1. Can Paul buy 6 kg of hamburger and 4 kg chicken if he wants to stay within his set budget? 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
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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. -1 -2 -3 -4 -5 1 2 3 4 5
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. -1 -2 -3 -4 -5 1 2 3 4 5
State the solution set.
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. Click here for the solution.
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.
x2 - 3x - 4 = 0 1. 1. 2.
x - 4
(x - 4)(x + 1)
x + 1
Use the boundary points to mark off test intervals on the number line.
x - 4
(x - 4)(x + 1)
x + 1
Determine the intervals when each of the factors is positive or negative. 3. 4.
Determine the solution using the number line.
Click here for the solution.

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

15. Graphing a Quadratic Inequality
Chapter
Graphing a Quadratic Inequality 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. The following pages contain solutions for the previous questions.
Click here to return to the start

18. Solutions Go back to the question. (0, 0) (-4, 0) (0, 4) (4, 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. Graph the boundary line for the inequality.
Solutions
Hamburger
Let h = kg of hamburger
c = kg of chicken
Write an inequality to represent the number of kilograms of each that Paul may purchase.
Chicken c h
Graph the boundary line for the inequality.
Go back to the question.

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

22. The solution set is {x | x < –3 or x > 4, x R}.
Solve the related equation to determine intervals of numbers that may be solutions of the inequality. -1 -2 -3 -4 -5 1 2 3 4 5
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. -1 -2 -3 -4 -5 1 2 3 4 5
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
The solution set is {x | x < –3 or x > 4, x R}.
State the solution set.
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. Use the boundary points to mark off test intervals on the number line.
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 = x = – 1
x2 - 3x - 4 = 0 1. 1. 2.
x - 4
(x - 4)(x + 1)
x + 1
x – 4
(x – 4)(x + 1)
x + 1 – +
Use the boundary points to mark off test intervals on the number line. 3.
Determine the intervals when each of the factors is positive or negative. 4.
x < – 1 or x > 4
Determine the solution using the number line.
Go back to the question.

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