Download Graphing Linear Inequalities in Two Variables

Digital Lesson
Graphing Linear
Inequalities in Two Variables
Expressions of the type x + 2y ≤ 8 and 3x – y > 6
are called linear inequalities in two variables.
A solution of a linear inequality in two variables is an
ordered pair (x, y) which makes the inequality true.
Example: (1, 3) is a solution to x + 2y ≤ 8
since (1) + 2(3) = 7 ≤ 8.
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The solution set, or feasible set, of a linear inequality
in two variables is the set of all solutions.
y
Example:
The solution set for x + 2y ≤ 8
is the shaded region.
2
2
x
The solution set is a half-plane. It consists of the line
x + 2y ≤ 8 and all the points below and to its left.
The line is called the boundary line of the half-plane.
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If the inequality is ≤ or ≥ ,
the boundary line is solid;
its points are solutions.
3x – y = 2
y
x
3x – y < 2
Example: The boundary line of the
3x – y > 2
solution set of 3x – y ≥ 2 is solid.
If the inequality is < or >,
the boundary line is dotted;
its points are not solutions.
Example: The boundary line of the
solution set of x + y < 2 is dotted.
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y
x
4
A test point can be selected to determine which side of
the half-plane to shade.
y
Example: For 2x – 3y ≤ 18
graph the boundary line.
(0, 0)
-2
2
x
The solution set is a half-plane.
Use (0, 0) as a test point.
(0, 0) is a solution. So all points on the (0, 0) side of the
boundary line are also solutions.
Shade above and to the left of the line.
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To graph the solution set for a linear inequality:
1. Graph the boundary line.
2. Select a test point, not on the boundary line, and
determine if it is a solution.
3. Shade a half-plane.
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Example: Graph the solution set for x – y > 2.
1. Graph the boundary line x – y = 2 as a dotted line.
y
(0, 0)
x
2. Select a test point not on
the line, say (0, 0).
(2, 0)
(0, -2)
(0) – 0 = 0 > 2 is false.
3. Since this is a not a solution, shade in the half-plane
not containing (0, 0).
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Solution sets for inequalities with only one variable
can be graphed in the same way.
y
Example: Graph the solution
set for x < -2.
4
x
-4
4
-4
y
Example: Graph the solution
set for x ≥ 4.
4
x
-4
4
-4
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A solution of a system of linear inequalities is an
ordered pair that satisfies all the inequalities.
x  y  8
Example: Find a solution for the system 
.
2 x  y  7
(5, 4) is a solution of x + y > 8.
(5, 4) is also a solution of 2x – y ≤ 7.
Since (5, 4) is a solution of both inequalities in the
system, it is a solution of the system.
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The set of all solutions of a system of linear inequalities
is called its solution set.
To graph the solution set for a system of linear
inequalities in two variables:
1. Shade the half-plane of solutions for each
inequality in the system.
2. Shade in the intersection of the half-planes.
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Example:
x  y  8
Graph the solution set for the system 
2 x  y  7
y
Graph the solution set for
x + y > 8.
Graph the solution set for
2x – y ≤ 7.
The intersection of these two
half-planes is the wedge-shaped
region at the top of the
diagram.
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2
2
x
11
Example: Graph the solution set for the system of
linear inequalities: 2 x  3 y  12

 2 x  3 y  6
y
-2x + 3y ≥ 6
Graph the two half-planes.
2
The two half-planes do not
intersect;
therefore, the solution set is
the empty set.
x
2
2x – 3y ≥ 12
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Example: Graph the solution set for the linear system.
(1) 2 x  3 y  3
(2) 
6 x  y  1

(3)  x  2
(4) 
 y  1
(2) y
(1)
4
x
-4
4
Graph each linear
inequality.
(4)
-4
(3)
The solution set is the intersection of all the half-planes.
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