2.8 Graph Linear Inequalities in Two Variables

 

 

Linear inequality in two variables

An inequality that can be written in one of the following forms:
Ax + By < C, Ax + By £ C, Ax + By > C, Ax + By ³ C

 

Solution of a linear inequality

An ordered pair (x, y) that makes the inequality true when the values of x and y are substituted into the inequality

 

Graph of a linear inequality

The set of all points in a coordinate plane that represent solutions of the inequality

 

Half-plane

The two regions of a coordinate plane that are separated by the boundary line of an inequality

 

Example 1

Checking solutions of inequalities

 

Check whether the ordered pairs (a) (3, 2) and (b) (-1, 4) are solutions of
4x + 2y > 6.

 

Ordered Pair

Substitute

Conclusion

a.  (3, 2)

4( 3 ) + 2( 2 )
= 16 > 6

(3, 2) _is_
a solution.

b.  (-1, 4)

4( -1 ) + 2( 4 )
= 4 > 6

(-1, 4) _is not_
a solution.

 

GRAPHING A LINEAR INEQUALITY

To graph a linear inequality in two variables, follow these steps:

 

Step 1 Graph the boundary line for the inequality. Use a _dashed_ line for < or > and a _solid_ line for £ or ³.

 

Step 2 Test a point _not on_ the boundary line to determine whether it is a solution of the inequality. If it _is_ a solution shade the half-plane containing the point. If it _is not_ a solution, shade the other half-plane.

 

 

 

 

 

Example 2

Graph a linear inequality with one variable

 

Graph y < -1 in a coordinate plane.

 

 


Solution

1.    Graph the boundary line y = -1. Use a _dashed_ line because the inequality symbol is <.

2.    Test the point (0, 0). Because (0, 0) _is not_ a solution of the inequality, shade the half-plane that _does not_ contain (0, 0).

Example 3

Graph a linear inequality with two variables

 

Graph 3x - 2y < -6 in a coordinate plane.

 

 


Solution

1.     Graph the boundary line 3x - 2y = -6. Use a _dashed_ line because the inequality symbol is <.

2.     Test the point (0, 0). Because (0, 0) _is not_ a solution of the inequality, shade the half-plane that _does not_ contain (0, 0).

It is often convenient to use (0, 0) as a test point. However, if (0, 0) lies on a boundary line, you must choose a different test point.
 
 

 


Example 4

Graph an absolute value inequality

 

Graph y > -3 ½x - 1½+ 2 in a coordinate plane.

 

 


1.         Graph the equation of the boundary, y = -3 ½x - 1½ + 2. Use a _dashed_ line because the inequality symbol is >.

2.         Test the point (0, 0). Because (0, 0) _is_ a solution of the inequality, shade the portion of the coordinate plane _outside_ the absolute value graph.