Graphing Linear Inequalities in Two Variables
Digital Lesson
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Expressions of the type x + 2y ≤ 8 and 3x – y > 6are 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.
Solution of Linear Inequalities
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The solution set, or feasible set, of a linear inequality in two variables is the set of all solutions.
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.
Example: The solution set for x + 2y ≤ 8 is the shaded region. x
y
2
2
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x
y
x
yIf the inequality is < or >, the boundary line is dotted; its points are not solutions.
If the inequality is ≤ or ≥ , the boundary line is solid; its points are solutions.
Example: The boundary line of the solution set of x + y < 2 is dotted.
Example: The boundary line of the solution set of 3x – y ≥ 2 is solid.
3x – y < 2
3x – y = 2
3x – y > 2
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x
y
Example: For 2x – 3y ≤ 18 graph the boundary line.
The solution set is a half-plane.
A test point can be selected to determine which side of the half-plane to shade.
Shade above and to the left of the line.
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.
(0, 0)
2-2
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To graph the solution set for a linear inequality:
2. Select a test point, not on the boundary line, and determine if it is a solution.
3. Shade a half-plane.
1. Graph the boundary line.
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x
y
Example: Graph the solution set for x – y > 2.
1. Graph the boundary line x – y = 2 as a dotted line.
2. Select a test point not on the line, say (0, 0).
(0) – 0 = 0 > 2 is false.
3. Since this is a not a solution, shade in the half-plane not containing (0, 0).
(0, 0)
(2, 0)
(0, -2)
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Solution sets for inequalities with only one variable can be graphed in the same way.
Example: Graph the solution set for x < - 2.
x
y
4
4
- 4
- 4
x
y
4
4
- 4
- 4
Example: Graph the solution set for x ≥ 4.
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A solution of a system of linear inequalities is an ordered pair that satisfies all the inequalities.
(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.
Example: Find a solution for the system .
72
8
yx
yx
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The set of all solutions of a system of linear inequalities is called its solution set.
1. Shade the half-plane of solutions for each inequality in the system.
To graph the solution set for a system of linear inequalities in two variables:
2. Shade in the intersection of the half-planes.
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x
yGraph the solution set for x + y > 8.
The intersection of these two half-planes is the wedge-shaped region at the top of the diagram.
Graph the solution set for 2x – y ≤ 7.
Example: Graph the solution set for the system
72
8
yx
yx
2
2
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Example: Graph the solution set for the system of
linear inequalities:
Graph the two half-planes.
The two half-planes do not intersect; therefore, the solution set is the empty set.
x
y
2x – 3y ≥ 12
-2x + 3y ≥ 6
632
1232
yx
yx
2
2
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x
y
4
4
- 4
- 4
Example: Graph the solution set for the linear system.
Graph each linear inequality.
The solution set is the intersection of all the half-planes.
1
2
16
332
y
x
yx
yx(1)(1)
(2)
(2)
(3)
(3)
(4)
(4)
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