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4.3 Systems of Linear InequalitiesGoal p Graph, write, and use a system of linear

inequalities.

VOCABULARY

System of linear inequalities in two variables Two or more inequalities in the same variables

Solution of a system of linear inequalities Any ordered pair (x, y) that is a solution of each inequality in the system

Graph of a system of linear inequalities The graph of all solutions of the system

Check whether (3, 22) is a solution of the system of inequalities.

a. x 1 y < 5 b. 4x 1 y 12x 8 2x 2 y < 29

SolutionORDERED PAIR SUBSTITUTE CONCLUSION

a. (3, 22) 3 1 ( 22 ) 5 1 < 5 (3, 22) is a 2( 3 ) 5 6 8 solution.

b. (3, 22) 4( 3 ) 1 ( 22 ) 5 10 1 (3, 22) is not 2 3 2 ( 22 ) 5 21 < 29 a solution.

Example 1 Check Solutions of Inequalities

1. 2y > 7 2. 2x 1 1 > 10 x 2 y < 10 3y 2 2 10

solution not a solution

Checkpoint Check whether (25, 4) is a solution of the system of inequalities.

Copyright McDougal Littell/Houghton Mifflin Company Lesson 4.3 Algebra 2 C&S Notetaking Guide 75

Your Notes

n2nt-04-NTG.indd 75n2nt-04-NTG.indd 75 10/4/07 3:16:46 AM10/4/07 3:16:46 AM

Your Notes GRAPHING A SYSTEM OF LINEAR INEQUALITIES

Step 1 Graph the boundary lines of the inequalities. Use a dashed line for an inequality with < or > . Use a solid line for an inequality with or .

Step 2 Shade the half-planes for the inequalities. The graph of the system is the region common to all the half-planes.

The double-shaded region shown above is the graph of x < 2 and y 1.


O

y

x1

1

y $ 1

x , 2

Example 2 Graph a System of Two Inequalities

Graph the system.

y < 2x 1 1 Inequality 1y 2x 2 3 Inequality 2

Solution

1. Graph the boundary line of each inequality. Use a dashed line for Inequality 1. Use a solid line for Inequality 2.

2. Shade the half-plane below y 5 2x 1 1 with horizontal lines to represent Inequality 1 .

Shade the half-plane on and below y 5 2x 2 3 with vertical lines to represent Inequality 2 .

The graph of the system is the overlap , or intersection, of the two shaded areas.

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y

x1

1

y , 2x 1 1y # 2x 2 3


Homework

Your Notes


3. y < 2x 2 3 4. x 21 x 1 3y 26 y 2 y > x 2 2

Checkpoint Graph the system.

O

y

x1

1

O

y

x1

1

Example 3 Graph a System of Three Inequalities

Graph the system.

x > 22 Inequality 1y 3 Inequality 223x 1 2y > 6 Inequality 3

Solution

The inequality x > 22 implies that the region is to the right of the dashed line x 5 22 .

The inequality y 3 implies that the region is on and below the line y 5 3 .

The inequality 23x 1 2y > 6 implies that the region is above the dashed line 23x 1 2y 5 6 .

The graph of the system is the shaded triangular region shown above.

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y

x1

1

Beginning here in Example 3, it is only necessary to show the solution region in your graphs of the systems of inequalities.


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