Unit 26Solving Inequalities

Presentation 1 Inequalities on a Number Line

Presentation 2 Solving Linear Inequalities

Presentation 3 Inequalities Involving Quadratic Terms

Presentation 4 Graphical Approach

Presentation 5 Linear Programming Problems

Unit 2626.1 Inequalities on a Number

Line

Illustrate these inequalities on the number line and list the integer values which they satisfy each one.

(a)

(b)

(c)

-3-4 0-1-2 31 2-5 4 65

-3-4 0-1-2 31 2-5 4 65

-3-4 0-1-2 31 2-5 4 65

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Which integer values of x satisfy all three inequalities? ?

Unit 2626.2 Solving Linear Inequalities

-3-4 0-1-2 31 2-5 4 65

Solve the following inequalities and illustrate each one on the number line

(a)

(b)

(c)

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-3-4 0-1-2 31 2-5 4 65

? ?

-3-4 0-1-2 31 2-5 4 65

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Unit 2626.3 Inequalities Involving

Quadratic Terms

Solve the following inequalities and illustrate each one on the number line.(a)

(b)

(c)

So either both andi.e. and which means that

Or both andi.e. andwhich means that

-3-4 0-1-2 31 2-5 4 65

-3-4 0-1-2 31 2-5 4 65

-3-4 0-1-2 31 2-5 4 65

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Unit 2626.4 Graphical Approach

A In the diagram below, find the three inequalities which define the shaded region

1 2 3 4 5-5 -4 -3 -2 -1 0

5

4

3

2

1

-1

-2

-3

-4

-5

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x

y

B Find the region satisfied by the inequalities

0 1 2 3 4 5 6 7 8 9

8

7

6

5

4

3

2

1

x

y

Unit 2626.5 Linear Programming

Problems

The shaded area in the diagram shows the solution of a set of inequalities in x and y. The variable x represents the number of boys in a cricket club and y represents the number of girls in the club.

(a) State, using arguments based on the graph, whether the cricket club can have as members(i) 10 boys and 5 girls (ii) 6 boys and 6 girlsSolution

(i) No, as point (10, 5) is not in the feasible region(ii) Yes, as point (6, 6) is in the feasible region ?

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0 5 10 15 20

20

15

10

5

x

y

(b) Write down the set of THREE inequalities that define the shaded region

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0 5 10 15 20

20

15

10

5

x

y

(c) A company sells unifroms for the club and makes a profit of $3.00 on a boys uniform and $5.00 on a girls uniform.

(i) Write an expression in x and y that represents the total profit made by the company on the sales of uniforms.

(ii) Calculate the minimum profit the company can make.

Solution

(i) ??(ii) Vertices of feasible region are

Corresponding values of P are

Minimum profit is at (1, 2) of value $13

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0 5 10 15 20

20

15

10

5

x

y