3-3 Systems of Inequalities

Quick Review

To solve a system of inequalities by graphing, first graph the boundaries for each inequality. Then shade the region(s) of the plane containing solutions valid for both inequalities.

Example

Solve the system of inequalities by graphing.

{ y > − 3 y ≤ − | x − 1 | left brace . table with 2 rows and 1 column , row1 column 1 , y greater than negative 3 , row2 column 1 , y less than or equal to negative vertical line x minus 1 vertical line , end table

Graph both inequalities and shade the region valid for both inequalities.

Exercises

Solve each system of inequalities by graphing.

15. { y < 4 x 3 x + y ≥ 5 left brace . table with 2 rows and 1 column , row1 column 1 , y less than 4 x , row2 column 1 , 3 x plus y greater than or equal to 5 , end table
16. { y < | 2 x − 4 | x + 5 y ≥ − 1 left brace . table with 2 rows and 1 column , row1 column 1 , y less than vertical line 2 x minus 4 vertical line , row2 column 1 , x plus 5 y greater than or equal to negative 1 , end table
17. { y ≤ | x + 2 | − 3 y ≥ 1 + 1 4 x left brace . table with 2 rows and 1 column , row1 column 1 , y less than or equal to vertical line x plus 2 vertical line negative 3 , row2 column 1 , y greater than or equal to 1 plus , 1 fourth , x , end table
18. { 2 x + 3 y > 6 x ≤ − 1 y ≥ 4 left brace . table with 3 rows and 1 column , row1 column 1 , 2 x plus 3 y greater than 6 , row2 column 1 , x less than or equal to negative 1 , row3 column 1 , y greater than or equal to 4 , end table
19. For a community breakfast there should be at least three times as much regular coffee as decaffeinated coffee. A total of ten gallons is sufficient for the breakfast. Write and graph a system of inequalities to model the problem.

3-4 Linear Programming

Quick Review

Linear programming is used to find a minimum or maximum of an objective function, given constraints as linear inequalities. The maximum or minimum occurs at a vertex of the feasible region, which contains the solutions to the system of constraints.

Example

Graph the system of constraints and name the vertices.

Objective function: P = 2x + y

{ x ≤ 8 y ≤ 5 x ≥ 0 , y ≥ 0 left brace . table with 3 rows and 1 column , row1 column 1 , x less than or equal to 8 , row2 column 1 , y less than or equal to 5 , row3 column 1 , x greater than or equal to 0 comma y greater than or equal to 0 , end table

Graph the inequalities and shade the area satisfying all inequalities.

The vertices of the feasible region are (0, 0), (0, 5), (8, 5), and (8, 0).

Evaluate the objective function at each vertex:

The maximum value occurs at (8, 5).

Exercises

Graph the system of constraints. Name the vertices. Then find the values of x and y that maximize or minimize the objective function.

20. { x ≥ 2 y ≥ 0 3 x + 2 y ≥ 12 left brace . table with 3 rows and 1 column , row1 column 1 , x greater than or equal to 2 , row2 column 1 , y greater than or equal to 0 , row3 column 1 , 3 x plus 2 y greater than or equal to 12 , end table

    Minimum for C = x + 5 y

21. { 3 x + 2 y ≤ 12 x + y ≤ 5 x ≥ 0 , y ≥ 0 left brace . table with 3 rows and 1 column , row1 column 1 , 3 x plus 2 y less than or equal to 12 , row2 column 1 , x plus y less than or equal to 5 , row3 column 1 , x greater than or equal to 0 comma y greater than or equal to 0 , end table

    Maximum for P = 3x + 5y

22. A lunch stand makes $.75 profit on each chef's salad and $1.20 profit on each Caesar salad. On a typical weekday, it sells between 40 and 60 chef's salads and between 35 and 50 Caesar salads. The total number sold has never exceeded 100 salads. How many of each type should be prepared in order to maximize profit?

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