1. Graphing Linear Inequalities in Two Variables
Digital Lesson
Graphing Linear Inequalities in Two Variables

2. Solution of Linear Inequalities
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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Solution of Linear Inequalities

3. Example: The solution set for x + 2y ≤ 8 is the shaded region.
The solution set, or feasible set, of a linear inequality in two variables is the set of all solutions.
Example: The solution set for x + 2y ≤ 8 is the shaded region. x y 2
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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Feasible Set

4. 3x – y = 2 x y
If the inequality is ≤ or ≥ , the boundary line is solid; its points are solutions.
3x – y < 2
Example: The boundary line of the solution set of 3x – y ≥ 2 is solid.
3x – 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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Boundary lines

5. Example: For 2x – 3y ≤ 18 graph the boundary line.
A test point can be selected to determine which side of the half-plane to shade. x y
Example: For 2x – 3y ≤ 18 graph the boundary line.
(0, 0) 2 -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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Test Point

6. Graphing an Inequality
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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Graphing an Inequality

7. Example: Graph an Inequality
Example: Graph the solution set for x – y > 2.
1. Graph the boundary line x – y = 2 as a dotted line. x y
(0, 0)
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).
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Graph an Inequality

8. Inequalities in One Variable
Solution sets for inequalities with only one variable can be graphed in the same way. x y 4
- 4
Example: Graph the solution set for x < - 2. x y 4
- 4
Example: Graph the solution set for x ≥ 4.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Inequalities in One Variable

9. Solution of a System of Linear Inequalities
A solution of a system of linear inequalities is an ordered pair that satisfies all the inequalities.
Example: Find a solution for the system
(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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Solution of a System of Linear Inequalities

10. 2. Shade in the intersection of the half-planes.
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.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Solution Set

11. Example: Graph a System of Two Inequalities
Example: Graph the solution set for the system x y
Graph the solution set for x + y > 8.
Graph the solution set for 2x – y ≤ 7. 2
The intersection of these two half-planes is the wedge-shaped region at the top of the diagram.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Graph a System of Two Inequalities

12. Example: Graph a System of Two Inequalities
Example: Graph the solution set for the system of linear inequalities: x 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.
2x – 3y ≥ 12
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Graph a System of Two Inequalities

13. Example: Graph a System of Four Inequalities
Example: Graph the solution set for the linear system.
(2) x y 4
- 4
(1)
(3)
(4)
Graph each linear inequality.
The solution set is the intersection of all the half-planes.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Graph a System of Four Inequalities