Lesson 7

Lesson 7: Graphing Linear and Quadratic Inequalities

 

Focus

 

A popular concept amongst urban planners is the development of “cities within cities.” These developments are designed to be one-stop shopping destinations as well as recreation and entertainment centres. More than your typical shopping mall, these complexes include restaurants, hotels, business offices, and even residential space.

 

The city-within-a-city concept has been particularly popular in Asian countries. In some Asian countries, booming economies and increasing consumer and housing demand have driven architects and designers to create all-encompassing mega-complexes.

Mathematics can help architects design complexes that meet the needs of residents and consumers alike.

 

In this lesson you will learn how you can use linear and quadratic inequalities to represent multiple variables, much like an architect might use mathematics to guide the development of a project involving multiple factors.

 

Outcomes

 

At the end of this lesson you will be able to

• explain, using examples, how test points can be used to determine the solution region that satisfies an inequality
  
  
• explain, using examples, when a solid or broken line should be used in the solution for an inequality
  
  
• sketch, with or without technology, the graph of a linear or quadratic inequality
  
  
• solve a problem that involves a linear or quadratic inequality

Lesson Questions

 

You will investigate the following questions:

• How are the properties of linear and quadratic inequalities in two variables related to their graphs?
  
  
• How are linear and quadratic inequalities in two variables applied in design?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 7 Assignment (Download the Lesson 7 Assignment and save it in your course folder now.)
  
  
• course folder submissions from Math Lab, Try This, and Share activities
  
  
• additions to Module 4 Glossary Terms and Formula Sheet
  
  
• work under Project Connection

Materials

 

You will need a ruler and graph paper.

 

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? This section, which includes Are You Ready? and Refresher, will help you find out.

 

Before beginning this lesson you should be able to

• solve linear inequalities in one variable
  
  
• graph linear functions
  
  
• identify slope and intercepts of a linear function
  
  
• graph a quadratic function

 

Are You Ready?

 

Complete these questions. If you need help, visit Refresher or contact your teacher.

1. Solve the following inequalities.
   
   
   1. 4x − 3 > 17 Answer
      
      
   2. 20 − 7x ≤ 8 + 5x Answer
      
      
2. For linear functions of the form y = mx + b, describe what m and b represent. Answer
   
   
3. Complete a table like the following.
   
   
    

   Function	Slope, if Linear	x-intercept	y-intercept	Sketch
   3x + 2y + 8 = 0
   y = 2x2 − 3x + 5

   
   
   Answer

How did the questions go? If you feel comfortable with the concepts covered in the questions, skip forward to Discover. If you experienced difficulties, use the resources in Refresher to review these important concepts before continuing through the lesson.

Refresher

 

Use the Inequality definition and applet to review the inequality symbols, the graphs of inequalities in one variable, and solving inequalities.

 

 

 

 

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Use Graphing y = mx + b to review how to graph linear functions in the form y = mx + b. Select Linear Functions, and then select Graphing Equations in the Form y = mx + b on Graph Paper.

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Refresh your skills in graphing a quadratic function.

• Return to Module 3: Lesson 3 to review how to convert a quadratic function in standard form to vertex form by completing the square.
  
  
• Use Quadratic Function (Vertex Form) to review the graphs of quadratic functions in the vertex form.
  
  
   
  
  
  

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

 

In this Discover section you will begin to explore how you can graph linear and quadratic inequalities in two variables.

 

Try This 1

 

Part A: Linear Inequalities

 

Launch Linear Inequalities in Two Variables - Activity A, and then follow the steps below. Record your responses as you work through the steps.

 

 

 

 

Step 1: Graph a linear function by adjusting the sliders for m and b. Notice that the line divides the coordinate plane into two areas. Record the linear function you graphed.

 

Step 2: Choose the “≤” symbol. How does the graph change?

 

Step 3: Select “Show solution test.” A blue point appears on the coordinate plane. What are the coordinates of the point? Does the test show this point as a solution to the inequality? On which side of the line does the point appear?

 

Step 4: Now drag the point to another part of the coordinate plane. Is this new point a solution to the inequality?

 

Step 5: Continue to experiment with different points, lines, and inequalities.

 

Analysis

1. How would you describe the location of the points that are solutions to the inequality?
   
   
2. A graph of a linear function would be only a straight line. Why do you think a shaded region is included in the graph of a linear inequality?
   
   
3. How does the graph of the line change when you change the inequality symbol from ≤ or ≥ to < or >?
   
   
4. What happens if you drag the point onto a dashed line? Is the point a solution? Likewise, what happens if you drag the point onto a solid line? Is the point a solution?
   
   
5. How would you instruct someone to sketch y > x + 3?
   
   
6. Describe the relationship between the inequality symbol of an inequality and the relative position of the shaded area on the corresponding graph.

Part B: Quadratic Inequalities

 

Launch Quadratic Inequalities in Two Variables - Activity A. Use your observations from Quadratic Inequalities in Two Variables to add to or revise your responses to Analysis questions 1 to 6. Then complete question 7.

7. What differences, if any, are there between linear inequalities and quadratic inequalities?

Save your responses in your course folder.

 

You will revisit these results in Explore.

Explore

 

The Linked Hybrid development is a twenty-first century housing complex located in Beijing, China. The complex is built on more than 15 ac (acres) of land and consists of over two million square feet dedicated to residential and commercial living spaces, entertainment and dining venues, underground parking, and even a kindergarten. This complex is truly a city within a city!

 

The design and construction of Linked Hybrid is an undertaking that requires consideration of a number of different factors. These factors are known as constraints. Constraints set limits, or boundaries, on what can be developed. Examples of constraints might be the space that is available for construction, the cost of materials and labour, and environmental considerations.

 

In this lesson you will explore linear and quadratic inequalities. You will learn how to use these inequalities to model situations involving constraints. You will explore multiple ways of graphing inequalities in two variables, with and without technology.

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Forms of Linear and Quadratic Inequalities

 

Linear and quadratic inequalities come in different forms. The following table summarizes the four different forms of each kind of inequality.

 

Linear Inequalities in Two Variables (a, b, and c are real numbers)	Quadratic Inequalities in Two Variables (a, b, and c are real numbers and a ≠ 0 )
ax + by < c	y < ax2 + bx + c
ax + by ≤ c	y ≤ ax2 + bx + c
ax + by > c	y > ax2 + bx + c
ax + by ≥ c	y ≥ ax2 + bx + c

 

An ordered pair (x, y) is considered a solution of an inequality if, when the values of x and y are substituted into the inequality, the inequality is true.

 

Try This 2

 

It is time to confirm the observations you made in Try This 1. You will focus on how the properties of linear and quadratic inequalities relate to their graphs.

 

Open Linear Inequalities in Two Variables - Activity A and Quadratic Inequalities in Two Variables - Activity A.

 

 

 

 

 

Consider the graphs shown.

 

1. For each graph, indicate whether the shaded region (in green) is above the boundary line or below the boundary line.
   
   
2. For each graph, use either Linear Inequalities in Two Variables or Quadratic Inequalities in Two Variables to reconstruct the graph. Record the inequality you used to represent each graph.

Save your response in your course folder.

 

Share 1

 

Compare your results from Try This 2 with a partner. Note that you may not have the same inequalities for all, or any, of the graphs. However, you should have identical inequality symbols.

1. Discuss or share any patterns you observed in both Try This 1 and Try This 2. These patterns may relate the inequality sign to
   
   
   • the properties of the boundary lines
     
     
   • the location of the shaded region
     
     
2. Select a point on the boundary line of one of the inequalities. Then select a second point in the shaded region with the same x-coordinate as the point you previously selected. How do the y-coordinates of these two points compare to the inequality symbol used in the inequality?
   
   
3. Study the graph shown.
   
   
    
   
   
   
   1. Is the shaded region above or below the boundary line?
      
      
   2. How does this change the pattern for describing the shaded region?

Save your response in your course folder.

Graphing Inequalities in Two Variables

 

The graph of a linear inequality begins with the graph of a boundary line. The boundary line divides the coordinate plane into two separate regions, known as (open) half-planes. The boundary line is either dashed or solid, depending on whether the inequality permits points on the line to be included.

 

The half-plane that contains the points that satisfy the inequality is shaded. This set of points is known as the solution region.

 

 

Consider this strategy for graphing an inequality in two variables:

 

Step 1: Decide whether your boundary should be dashed or solid. If the inequality symbol is < or >, the boundary should be dashed. If the inequality symbol is ≤ or ≥, the boundary should be solid.

 

Step 2: Graph the boundary or the function corresponding to the inequality.

 

Step 3: Choose a point not on the boundary line. This is the test point.

 

Step 4: Check whether the test-point coordinates satisfy the inequality. If they do, shade the half-plane containing the point. If they don’t, shade the half-plane that does not contain the point.

 

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Try This 3

 

Part A

1. Graph the linear inequality 2x + 3y ≤ 6 using the steps in the strategy you just learned for graphing inequalities in two variables.

Writing an Inequality to Describe a Given Graph

 

You have learned how to graph linear and quadratic inequalities in two variables. You will now learn how to write an inequality to describe a given graph.

 

Try This 4

1.  
   1. For the graph shown, write the equation of the boundary.
      
      
   2. What inequality sign should be used? Write the inequality.
      
      
      
2. In the graph shown, the equation of the boundary is given. Replace the equal sign in the equation 2x − 3y = 6 with an inequality symbol such that the resulting inequality describes the graph.
   
   
    
   
   

Save your responses in your course folder.

 

Share 2

 

With a classmate, compare your strategies for each of the exercises in Try This 4. If both you and your partner used the same strategy, see if you can develop a second strategy together. Discuss the relative benefits and drawbacks of each strategy.

Using a Test Point to Determine the Graph of an Inequality

 

Watch Using a Test Point to see one way to approach Try This 4, question 2 using a test point. In the video, a misconception is introduced. Pay attention to what the misconception is and how to avoid it.

 

 

 

 

 

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Self-Check 2

 

Test your skills by completing Understanding Inequalities.

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Working with Constraints

 

Linear and quadratic inequalities can be used to model situations with constraints. For example, an architect may be constrained by such factors as the available space, the cost of the project, and environmental considerations.

 

Launch Graphing Linear Inequalities. In the lower right corner of the window, click on the green bar showing the page number. Select page 9, and then complete the exercises on pages 9 to 12. What constraints are imposed in this example situation?

 

 

 

 

 

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Connect

 

Lesson 7 Assignment

Lesson 7 Summary

 

In this lesson you investigated the following questions:

• How are the properties of linear and quadratic inequalities in two variables related to their graphs?

• How are linear and quadratic inequalities in two variables applied in design?

You explored linear inequalities and quadratic inequalities in two variables, and you learned how to graph these inequalities. To graph a linear or quadratic inequality, you must answer two key questions:

• How do I graph the boundary?

• How do I know which half-plane should be shaded?

The following table summarizes the properties of the graph of linear and quadratic inequalities given each type of inequality symbol.

 

	<	≤	>	≥
Boundary Type	dashed	solid	dashed	solid
Shading Relative to Boundary when y Is Isolated	below	below	above	above
Linear Inequality
Quadratic Inequality

 

You also learned how to use the graphs of linear and quadratic inequalities to model problems involving constraints. These models are used in design and other industries to find possible solutions to problems.

 

In the final lesson of this module you will investigate solutions to quadratic inequalities in one variable. You will investigate both graphical and algebraic methods as solution strategies.