7-3 systems of inequalities (presentation)

8/7/2019 7-3 Systems of Inequalities (Presentation)

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7-3 Systems of Inequalities

Unit 7 Systems and Matrices



Concepts and Objectives Systems of Inequalities (Obj. #25)

Graph systems of inequalities Identify solutions to systems of inequalities

Use linear programming to identify maximums and

m n mums



Graphing Review To graph a linear inequality, put the inequality into

slope-intercept form. Plot the y -intercept and count the slope from there (rise

over run)

Symbol Line Shade< below

> above

≤ below

≥ above



Graphing Review Example: Graph the solution to + >2 3 6x y

> − +3 2 6y x

> − +2

23

y x



Graphing Review Example: What is the solution to

Vertex: (0, –2)

≥ −22y x



Systems of Inequalities The solution to a system of inequalities will be the graph

of the overlap between the two (or more) inequalities. You must graph the system to show the solution.

If the inequalities do not overlap, then there is no .



Systems of Inequalities Example: What is the solution to the system?

− > −+ ≤

2 4

3 2

x y

x y

Inequality #1: − > − −2 4y x

< +1

22

y x



Systems of Inequalities Example: What is the solution to the system?

− > −+ ≤

2 4

3 2

x y

x y

Inequality #2: ≤ − +3 2y x



Linear Programming An important application of mathematics is called linear

programming. We use linear programming to find anoptimum value.

To solve a linear programming problem:

.

Write all of the relevant inequalities. Theseconstraints form the system of inequalities.

Define the function to be maximized/minimized,

which is called the objective function. The solution, if it exists, occurs at a vertex of the

region of feasible solutions.



Linear Programming Example: An office manager wants to buy some filing

cabinets. He knows that cabinet A costs $10 each,requires 6 sq. ft. of floor space, and holds 8 cu. ft. of files.Cabinet B costs $20 each, requires 8 sq. ft. of floor space,and holds 12 cu. ft. of files. He can s end no more than

$140 due to budget limitations, and his office has roomfor no more than 72 sq. ft. of cabinets. He wants tomaximize storage capacity within the limits imposed byfunds and space. How many of each type of cabinet

should he buy?



Linear Programming Example (cont.):

Let x = # of cabinet A unitsy = # of cabinet B units

Inequality #1: (cost)

Inequality #2:

Also, neither x nor y can be negative

+ ≤10 20 140x y

+ ≤6 8 72y



Linear Programming Example (cont.):

≤ − + ≤ − +

1 72

39

y x

x

The optimal number of cabinets isat the vertex of the system: (8, 3)

≥ ≥

4

0

0

y

x

8 Cabinet A and 3 Cabinet B will store 100 cu. ft. of files.



Homework College Algebra (brown book)

Page 905: 30, 34-36, 39, 42, 59-62, 77, 80 Turn in: 30, 34, 42, 60, 61, 80

7-3 systems of inequalities (presentation)

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