5.2Systems of Linear Inequalities in Two Vari- ables

5.2 Systems of Linear Inequalities in Two Variables

In the same way as we combined multiple equations in a system of equations, we can combine multiple inequalities to get a system of inequalities. For example,

2x + y 2 7x - 3y 5 -x + y > 1

x 0 y< 1

Graph of a System of Linear Inequalities

Definition. Solution set of a system of linear inequalities in two variables is the set of all ordered pairs of real numbers (x, y) that simultaneously satisfy all the inequalities in the system. A graph of all such ordered pairs is called the solution region for the system. To solve a system of inequalities graphically means to graph its solution region.

To simplify the discussion that follows, we consider only systems of linear inequalities where equality is included in each statement in the system.

Ch 5. Lin. Ineq. and Lin. Prog.

5.2 Systems of Linear Inequalities in Two Variables

Example 1 Solve the following system of linear inequalities graphically:

3x+ y 9 x-2y 0

y

10 9 8 7 6 5 4 3 2 1

-10-9 -8 -7 -6 -5 -4 -3 -2 --11 -2 -3 -4 -5 -6 -7 -8 -9 -10

x 1 2 3 4 5 6 7 8 9 10

The points of intersection of the lines that form the boundary of a solution region will play a fundamental role in the solution of linear programming problems, which are discussed in the next section.

Definition (Corner Point) A corner point of a solution region is a point in the solution region that is the intersection of two boundary lines.

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Ch 5. Lin. Ineq. and Lin. Prog.

5.2 Systems of Linear Inequalities in Two Variables

Example 2 Solve the following system of linear inequalities graphically and find the corner points:

2x+ y 12 x+ y 6 x+2y 11

y

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

-15-14-13-12-11-10-9 -8 -7 -6 -5 -4 -3 -2 --11 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15

x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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Ch 5. Lin. Ineq. and Lin. Prog.

5.2 Systems of Linear Inequalities in Two Variables

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Ch 5. Lin. Ineq. and Lin. Prog.

5.2 Systems of Linear Inequalities in Two Variables

Definition (Bounded and Unbounded Solution Regions) A solution region of a system of linear inequalities is bounded if it can be enclosed within a circle. If it cannot be enclosed within a circle, it is unbounded.

For example, on the graphs below, the solution region (a) is bounded, and the solution region (b) is unbounded. This definition will be important in the next section.

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