Digital Lesson

Expressions of the type x + 2 y
≤ 8 and 3 x
– 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 + 2 y
≤ 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 + 2 y
≤ 8 is the shaded region.
2
2 x
The solution set is a half-plane . It consists of the line x + 2 y
≤ 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.
3 x
– y < 2
Example: The boundary line of the solution set of 3 x – y ≥ 2 is solid.
y
3 x
– y = 2
3 x
– y > 2 x y
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.
x
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A test point can be selected to determine which side of the half-plane to shade.
y
Example : For 2 x – 3 y ≤ 18 graph the boundary line.
(0, 0)
2 x
-2
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.
(0, 0) y 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
Example : Graph the solution set for x ≥ 4.
- 4 y
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.
Example x

2 x
 y y

8
: Find a solution for the system .
7
(5, 4) is a solution of x + y > 8.
(5, 4) is also a solution of 2 x
– 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

2 x
 y y

8
Graph the solution set for the system
7 y
Graph the solution set for x + y > 8.
Graph the solution set for
2 x
– y
≤ 7.
2
2
The intersection of these two half-planes is the wedge-shaped region at the top of the diagram.
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x
11

Example : Graph the solution set for the system of linear inequalities:

2

  x
2
 x
3 y


3 y
12

6 y
-2 x + 3 y ≥ 6
Graph the two half-planes.
2
The two half-planes do not intersect; therefore, the solution set is the empty set .
2
2 x – 3 y ≥ 12 x
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Example : Graph the solution set for the linear system.
(1)
(2)


2
6 x x


3 y y


3

1
(3)
(4)

 x y



2

1
(1)
(2)
4 y x
- 4 4
(4)
Graph each linear inequality.
- 4
(3)
The solution set is the intersection of all the half-planes.
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