1. Graphing Linear Inequalities in Two Variables
GOAL: graph a linear inequality 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.
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Solution of Linear Inequalities

3. Step 1: Put into slope intercept form
Using What We Know
Sketch a graph of x + y < 3
Step 1: Put into slope intercept form
y <-x + 3
Step 2: Graph the line
y = -x + 3

4. Less than means to the
left or below.
To check it, pick any point that is not on the line. (0,0) is an easy point to use.
x + y < 3
Substitute 0 for x and y.
0 + 0 < 3
0 < 3
Decide if this is true or false.
Is 0 less than 3?
If it is true, you shade on the same side of the line of the point you picked.
If it is false, you shade on the opposite side of the line where the point you picked lies.

5. Graphing Linear Inequalities
The graph of a linear inequality in two variables is the graph of all solutions of the inequality.
The boundary line of the inequality divides the coordinate plane into two half-planes: a shaded region which contains the points that are solutions of the inequality, and an unshaded region which contains the points that are not.
GRAPHING A LINEAR INEQUALITY
The graph of a linear inequality in two variables is a half-plane. To graph a linear inequality, follow these steps. 1
STEP
Graph the boundary line of the inequality. Use a dashed line for < or > and a solid line for £ or ³. 2
STEP
To decide which side of the boundary line to shade, test a point not on the boundary line to see whether it is a solution of the inequality. Then shade the appropriate half-plane.

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.
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Graphing an Inequality

7. Graphing a Linear Inequality
Sketch a graph of y  3

8. Graphing an Inequality in Two Variables
Graph x < 2
Step 1: Start by graphing the line x = 2
Now what points would give you less than 2?
Since it has to be x < 2 we shade everything to the left of the line.

9. Graph a) y < –2 and b) x £ 1 in a coordinate plane.
SOLUTION
Graph the boundary line x = 1. Use a solid line because x £ 1.
Graph the boundary line y = –2. Use a dashed line because y < – 2.
Test the point (0, 0).
Because (0, 0) is a solution of the inequality, shade the half-plane to the left of the line.
Test the point (0, 0).
Because (0, 0) is not a solution of the inequality, shade the half-plane below the line.

10. Some Helpful Hints If the sign is > or < the line is dashed
If the sign is  or  the line will be solid
When dealing with just x and y.
If the sign > or  the shading either goes up or to the right
If the sign is < or  the shading either goes down or to the left

11. When dealing with slanted lines
If it is > or  then you shade above
If it is < or  then you shade below the line

12. 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.
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Solution Set

13. 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

14. Example 1 Which ordered pair is a solution of 5x - 2y ≤ 6? (0, -3)
(5, 5)
(1, -2)
(3, 3)
ANSWER: A. (0, -3)

15. Example 2 Graph the inequality x ≤ 4 in a coordinate plane.
HINT: Remember HOY VEX.
Decide whether to
use a solid or
dashed line.
Use (0, 0) as a
test point.
Shade where the
solutions will be. y x 5 -5

16. Example 3 Graph 3x - 4y > 12 in a coordinate plane.
Sketch the boundary line of the graph.
Find the x- and
y-intercepts and
plot them.
Solid or dashed
line?
Use (0, 0) as a
test point.
Shade where the
solutions are. y x 5 -5

17. Example 4: Using a new Test Point
Graph y < 2/5x in a coordinate plane.
Sketch the boundary line of the graph.
Find the x- and y-intercept and plot them.
Both are the origin!
Use the line’s slope
to graph another point.
Solid or dashed
line?
Use a test point
OTHER than the
origin.
Shade where the
solutions are. y x 5 -5

18. Homework