2.8 Graph Linear Inequalities in Two Variables
Goal  Graph linear inequalities in two variables.
Your Notes
VOCABULARY
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  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.
Checkpoint Check whether the ordered pair is a solution of 2x  y  8.
1. (6, 2)
is not a solution
2. (3, 1)
is a solution
Your Notes
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 .
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.
Your Notes
Example 4
Graph an absolute value inequality
Graph y > 3 x  1+ 2 in a coordinate plane.
1.
Graph the equation of the boundary, y = 3 x  1 + 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.
Checkpoint Graph the inequality in a coordinate plane.
3.
x < 2
4.
y  x + 2
5.
9x + 3y > 9
6.
y  2 |x + 2|  l
Homework
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