What you should learn

To graph inequalities in the coordinate plane

NCTM Curriculm Standards 2, 6 - 10

 

Introduction: Rapid Cycle, Inc. is a manufacturer and distributor of racing bicycles. It takes 3 hours to assemble a bicycle and 1 hour to road test a bicycle. Each technician in the company works no more than 45 hours a week. How many racing bikes can one technican assemble, and how many can he or she road test in one week?

Let x represent the number of bikes that are assembled in a week, and let y represent the nuber of bikes that are road-tested. Then the following inequality can be used to represent the solution.

3x + y 45

There are an infinite number of ordered pairs that are solutions to this inequality. The easiest way to show all of these solutions is to draw a graph of the inequality. Before doing this, let's consider some simpler inequalities.

Exercise 1: From the set {(3, 4), (0, 1), (1, 4), (1, 1)}, which ordered pairs are part of the solution set for 4x + 2y < 8?

Let's use a table to substitute the x and y values of each ordered pair into the inequality.

The ordered pairs {(0, 1), (1, 1)} are part of the solution set of 4x + 2y < 8. The graph above shows the four ordered pairs of the replacement set and the equation 4x + 2y = 8. Notice the location of the two ordered pairs that are solutions for 4x + 2y < 8 in relation to the graph of the line.

Activity: Exploration Programming

You can use the following graphing calculator program to find out if a given ordered pair (x, y) is a solution for the inequality 5x - 3y 15.

PROGRAM: XYTEST

:Disp "IS (X, Y) A " , "SOLUTION?"

:Prompt X, Y

: If 5X-3Y15

:Then

:Disp "YES"

:Else

:Disp "NO"

Your Turn:

a. Try the program for ten ordered pairs (x, y). Keep a list of which ordered pairs you tried and which ones were solutions.

b. How do you think you could change this program to test the inequalityy 2x + y 2y?

c. Use your changed program to find the solution set if x = {-1, 0, 1} and y = { -2, -1, 0, 1}.

 

 

The solution set for an inequality contains many ordered pairs when the domain and range are the set of real numbers. The graphs of all of these ordered pairs fill an area on the coordinate plane called a half-plane. An equation defines the boundary or edge for each half-plane. For example, suppose you wanted to graph the inequality y > 5 on the coordinate plane.

First determine the boundary by graphing y = 5.

Since the inequality involves only >, the line should be dashed. The boundary divides the coordinate plane into two half planes.

To determine which half-plane contains the solution, choose a point from each half-plane and test it in the inequality.

Try (7, 10).

y > 5

10 > 5 true

Try (4, 0).

y > 5

0 > 5 false

The half-plane that contains (7, 10) contains the solution. Shade that half-plane.

Exercise 2: Graph y + 2x3

First solve for y in terms of x

y + 2x 3

y + 2x - 2x 3 - 2x

y 3 - 2x

 

Graph y = 3 - 2x. Since y 3 - 2x means y < 3 - 2x or y = 3 - 2x, the boundary is included in the graph and should be drawn as a solid line.

 

Select a point in one of the half-planes and test it. For example, use the origin (0, 0).

 

The half plane that contains the origin should be shaded.

Check: Test a point in the other half-plane, for example (3, 3).

 

Since the statement is false, the half-plane containing (3, 3) is not part of the solution.

 

 

When solving real-life inequalities, the domain and range of the inequality are often restricted to nonnegative numbers or whole numbers.

Exercise 3: Refer to the application at the beginning of the lesson. How many racing bikes will the technician be able to assemble and road test?

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 15 - 45 odd, 46, 47, 49 - 56

Alternative Homework: Enriched: 16 - 46 even, 47 - 56

Extra Practice: Students book page 774 Lesson 7-8

Extra Practice Worksheet: Click Here.

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