Pre-Publication - 11...graph paper, ruler, and coloured pencils 6.1 EXPLORE • For...

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Transcript of Pre-Publication - 11...graph paper, ruler, and coloured pencils 6.1 EXPLORE • For...

NEL290

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LEARNING GOALS

You will be able to develop algebraic and graphical reasoning by

Modelling and solving problems algebraically and graphically using linear inequalities in two variables

Modelling and solving problems algebraically and graphically using systems of linear inequalities in two variables

Solving optimization problems using linear programming

How can linear inequalities help shoppers decide what and how much to buy?

6Chapter

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Systems of Linear

Inequalities

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NEL292 Chapter 6 Systems of Linear Inequalities

6 Getting StartedBalancing Two Part-Time Jobs

Marnie has two part-time jobs. She earns minimum wage working at the information desk in a hospital and $4 above minimum wage helping with her mothers house-cleaning business. Marnie works in whole-hour increments only. She enjoys the work at the hospital more than house cleaning and does not work more than 15 h a week, since she often has a lot of homework and she plays on her schools volleyball team.

How many hours can Marnie work at the hospital and earn at least $160 a week?

A. Predict the solution to the problem. Explain your prediction to a partner.

B. If Marnie works 5 h at the hospital, how many hours will she work for her mother? Will she earn at least $160? Explain.

C. If Marnie works 10 h at the hospital, how many hours will she work for her mother? Will she earn at least $160? Explain.

D. What type of linear inequality is being described in the problem? Justify your choice.

E. Use a variable to represent the number of hours that Marnie works at the hospital. Use the same variable to create an expression for the number of hours that she works for her mother.

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linear inequality

A linear inequality is a relationship between two linear expressions in which one expression is less than (,), greater than (.), less than or equal to (#), or greater than or equal to ($) the other expression.

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NEL 293Getting Started

F. To what set of numbersnatural, whole, integer, rational, or realdoes the variable you defined in part E belong? Justify your decision.

G. Write a linear inequality to represent Marnies situation. Explain why your linear inequality makes sense.

H. Solve the linear inequality, and represent the solution on a number line.

I. Describe how your prediction in part A compares with your solution in part H.

WHAT DO You Think?Decide whether you agree or disagree with each statement. Explain your decision.

1. The solution to a linear inequality can be modelled graphically on a number line.

2. A linear inequality can be modelled using a solid line or dashed line.3. The same algebraic methods that are used to solve a linear equation

can be used to solve a linear inequality.4. A linear equation has been graphed below. Its solutions can only be

represented by points with integer coordinates, such as (1, 5) and (21, 1).

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NEL294 Chapter 6 Systems of Linear Inequalities

Graphing Linear Inequalities in Two Variables

Solve problems by modelling linear inequalities in two variables.

INVESTIGATE the MathAmir owns a health-food store. He is making a mixture of nuts and raisins to sell in bulk. His supplier charges $25/kg for nuts and $8/kg for raisins.

What quantities of nuts and raisins can Amir mix together if he wants to spend less than $200 to make the mixture?

A. Suppose that Amir wants to spend exactly $200 to make the mixture. Work with a partner to create an equation that represents this situation.

B. To what set of numbers does the domain and range of the two variables in your equation belong? Use this information to help you graph the equation on a coordinate plane.

C. Explain why the graph is a line segment, not a ray or a line.

D. What region of the coordinate plane includes points representing quantities of nuts and raisins that Amir could use if he wants to spend less than $200? How do you know?

E. There are many possible solutions to Amirs problem. Plot at least three points that represent reasonable solutions to Amirs problem. Explain why you chose these points.

Reflecting

F. Discuss and then decide whether the solution set for Amirs problem is represented by i) points in the region above the line segment.ii) points in the region below the line segment.iii) points on the line segment.

G. Why might the line segment be considered a boundary of the solution set?

H. Why might you use a dashed line segment for this graph instead of a solid line segment?

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solution set

The set of all possible solutions.

YOU WILL NEED

graphing technology OR graph paper, ruler, and coloured pencils

6.1

EXPLORE

Forwhichinequalitiesis(3,1)a possible solution? How do you know?

a) 1323x . 4y b) 2y 2 5 # x c) y 1 x ,10 d) y $ 9

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NEL 2956.1 Graphing Linear Inequalities in Two Variables

I knew that the graph of the linear equation 22x 1 5y 510 would form the boundary of the linear inequality 22x 1 5y $10.

The domain and range are not stated and no context is given, so I assumed that the domain and range are the set of real numbers. This means that the solution set is continuous . continuous

A connected set of numbers. In a continuous set, there is always another number between any two given numbers. Continuous variables represent things that can be measured, such as time.

I knew that I needed to plot and join only two points to graph the linear equation. I decided to plot the two intercepts.

To determine the y-intercept, Isubstituted0forx.

To determine the x-intercept, Isubstituted0fory.

APPLY the Mathexample 1 Solving a linear inequality graphically

when it has a continuous solution set in two variables

Graph the solution set for this linear inequality: 22x 1 5y $ 10

Roberts Solution: Using graph paper

Linear equation that represents the boundary:22x 1 5y 5 10

The variables represent numbers from the set of real numbers.x [ R and y [ R

y-intercept: 22x 1 5y 5 1022(0) 1 5y 5 10

5y5

5105

y 5 2The y-intercept is at (0, 2).

x-intercept: 22x 1 5y 5 1022x 1 5(0) 5 10

22x22

51022

x 5 25The x-intercept is at (25, 0).Pr

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NEL296 Chapter 6 Systems of Linear Inequalities

Since my test point below the boundary was not a solution, I shaded the half plane that did not include my test point. This was the region above the boundary.

I used green shading to show that the solution set belongs to the set of real numbers.

Since the domain and range are in the set of real numbers, I knew that the solution set is continuous. Therefore, the solution region includes all points in the shaded area and on the solid boundary.

I needed to know which half plane , above or below

the boundary, represents the solution region for the linear inequality.

To find out, I substituted the coordinates of a point in the half plane below the line. I used (0,0)becauseitmadethecalculations simple.

I already knew that the solution region includes points on the boundary, so I didnt need to check a point on the line.

half plane

The region on one side of the graph of a linear relation on a Cartesian plane.

Since the linear inequality has the possibility of equality ($), and the variables represent real numbers, I knew that the solution region includes

all the points on its boundary. Thats why I drew a solid green line through the intercepts.

solution region

The part of the graph of a linear inequality that represents the solution set; the solution region includes points on its boundary if the inequality has the possibility of equality.

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(0, 2)(5, 0)

Test (0, 0) in 22x 1 5y $ 10.LS RS22(0) 1 5(0) 100

Since 0 is not greater than or equal to 10, (0, 0) is not in the solution region.

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2x 5y 10If the solution set to a linear inequality is continuous and the sign includes equality (# or $), a solid green line is used for the boundary, and the solution region is shaded green, as shown to the right.

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NEL 2976.1 Graphing Linear Inequalities in Two Variables

The domain and range are not stated and no context is given, so I assumed the domain and range are in the set of real numbers. This means that the solution set is continuous.

To enter the linear inequality into my graphing calculator, I had to isolate y.

I didnt have to reverse the inequality sign because I didnt divide or multiply by a negative value.

I entered the inequality into the calculator and graphed it.

I knew that the boundary was correctly shown because a solid line means equality is possible.

To verify whether the correct half plane was shaded, I used (1,4)inthehalfplaneabove the boundary as a tes