Transcript Chapter 9 FINAL - McGraw

McGraw-Hill Ryerson
Pre-Calculus 11
Chapter 9
Linear and Quadratic Inequalities
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McGraw-Hill Ryerson Pre-Calculus 11
Teacher Notes
1. This lesson is designed to help students conceptualize the
main ideas of Chapter 9.
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Chapter
9
Linear Inequalities
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
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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.
3

2,


2 

 3 
  2 ,2 


3


2,


2 

 0,0 
0,4
0, 4
 4,0
 4,0
x  4y  4
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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Click here for the solution.
Chapter
9
Graphing Linear Inequalities
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
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Shade
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.
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Chapter
9
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Graphing a Linear Inequality
Match each inequality to its graph.
Then, click on the graph to check the answer.
Linear Inequalities
Chapter
9
Write an inequality that represents each graph.
2.
1.
(2, 4)
(0, 3)
0
0
(2, -1)
2x  y  3
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(0, -2)
3x  y  2
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
Let h = kg of hamburger
kilograms of each that Paul may purchase.
c = kg of chicken
Chicken
Write the equation of the boundary line
below and draw its graph.
Shade the solution region for the inequality.
Hamburger
Click here for the solution.
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Solve an Inequality
Chapter
c
9
Chicken
5h  6.5c  48
h
Hamburger
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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Click here for the solution.
Chapter
9
Quadratic Inequalities
2
Solve x – 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
-5
-4
-4
-3
-3
-2
-2
-1
-1
0
0
1
1
2
2
3
4
5
3
4
5
State the solution set.
Click here for the solution.
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Chapter
9
Quadratic Inequalities
2
Solve x – 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.
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Chapter
9
1.
Sign Analysis
2
Solve x – 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.
x-4
Determine the intervals when each of
the factors is positive or negative.
x+1
(x - 4)(x + 1)
Determine the solution using the
number line.
Click here for the solution.
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Chapter
9
1.
Sign Analysis
2
Solve x – 3x – 4 > 0.
Use the pen tool to create a graph of
the related function to confirm your
solutions.
Click here for the solution.
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Chapter
9
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Graphing a Quadratic Inequality
Choose the correct shaded region to complete the graph of the inequality.
Circle your choice using the pen tool.
Chapter
9
Quadratic Inequality in Two Variables
Match each inequality to its graph using the pen tool.
y  2x 2 , y  2x 2 , y   x 2
y   x 2  4 x  6, y  x 2  6 x  10, y  x 2  6 x  5
y  2x 2
y   x 2  4x  6
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y  x 2  6x  5
y  2x 2
y  x 2  6x  10
y  x 2
The following pages contain solutions for the
previous questions.
Click here to return to the start
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Solutions
(0, 4)
0
(-4, 0)
(0, 0)
(4, 0)
(0, -4)
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Go back to the question.
Solutions
An example method for graphing an inequality would be:
1
• Slope of the line is 3 .
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.
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• 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.
Go back to the question.
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.
c
Chicken
Graph the boundary line for the inequality.
h
Hamburger
Go back to the question.
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Solutions
c
5h  6.5c  48
Chicken
(0, 7.38)
(3, 5)
(6, 4)
h
Hamburger
1. Can Paul buy 6 kg of hamburger and 4 kg chicken if he wants to stay within his set budget?
The point (6, 4) is not within the shaded region. Paul could
not purchase 6 kg of hamburger and 4 kg of chicken.
2. How many kilograms of chicken can Paul buy if he decides not to buy any hamburger?
This is the point (0, 7.38). Buying no hamburger would be
the y-intercept of the graph.
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.
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Go back to the question.
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.
State the solution set.
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-5
-4
-3
-2
-1
-4
2
3
-3
-2
-1
0
1
2
5
4
Test point 5:
2
(5) - (5) - 12 > 0
True
Test point 0:
2
(0) - (0) - 12 > 0
False
Test point -5:
2
(-5) - (-5) - 12 > 0
True
-5
1
0
3
4
5
The solution set is {x | x < –3 or x > 4, x R}.
Go back to the question.
Solutions
2
Solve x – 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.
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Solutions
1.
2
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.
x – 3x – 4 = 0
(x – 4)(x + 1) = 0
x – 4 = 0 or x + 1 = 0
x=4
x=–1
–
–
+
–
+
+
+
–
+
x–4
x+1
Determine the intervals when each of
the factors is positive or negative.
Determine the solution using the
number line.
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(x – 4)(x + 1)
x < – 1 or x > 4
Go back to the question.
Solutions
1.
A graph of the related function may
be used to confirm your solutions.
x < – 1 or x > 4
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Go back to the question.