1. Graphing Linear Inequalities in Two Variables (5-6)
Objective: Graph linear inequalities on the coordinate plane. Solve inequalities by graphing.

2. Graph Linear Inequalities
The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality.
An equation defines a boundary, which divides the coordinate plane into two half-planes.
The boundary may or may not be included in the graph.
When it is included, the solution is a closed half-plane.
When not included, the solution is an open half-plane.

3. Graph Linear Inequalities
Use the following steps to graph a linear inequality.
Graph the boundary. Use a solid line when the inequality contains ≤ or ≥. Use a dashed line when the inequality contains < or >.
Use a test point to determine which half-plane should be shaded.
Shade the half-plane that contains the solution.

4. Example 1 Graph 2y – 4x > 6. 2y – 4x > 6 +4x +4x 2y > 4x + 6
Graph the line y = 2x + 3 using a dashed line.
2y – 4x > 6
+4x +4x
2y > 4x + 6
y > 2x + 3
Test Point: (0, 0)
0 > 2(0) + 3
0 > 3
False

5. Check Your Progress Choose the best answer for the following.
Graph y – 3x < 2.
y – 3x < 2 A. B.
+3x +3x
y < 3x + 2
Graph the line y = 3x + 2 using a dashed line.
Test Point: (0, 0) C. D.
0 < 3(0) + 2
0 < 2
True

6. Example 2 Graph x + 4y ≥ 2. x + 4y ≥ 2 -x -x 4y ≥ -x + 2 4 4 4
Graph the line y = -¼x + ½ using a solid line.
x + 4y ≥ 2
-x x
4y ≥ -x + 2
y ≥ -¼x + ½
Test Point: (0, 0)
0 ≥ -¼(0) + ½
0 ≥ ½
False

7. Check Your Progress Choose the best answer for the following.
Graph x + 2y ≥ 6.
x + 2y ≥ 6 A. B.
-x x
2y ≥ -x + 6
y ≥ -½x + 3
Graph the line y = -½x + 3 using a solid line. C. D.
Test Point: (0, 0)
0 ≥ -½(0) + 3
0 ≥ 3
False

8. Example 3
We can use a coordinate plane to solve inequalities with one variable.
Use a graph to solve 2x + 3 ≤ 7.
2x + 3 ≤ 7
2x ≤ 4
x ≤ 2

9. Check Your Progress Choose the best answer for the following.
Use a graph to solve 5x – 3 > 17.
x > 20
x > 3
x < -4
x > 4
5x – 3 > 17
5x > 20

10. Example 4
When using inequalities to solve real-world problems, the domain and the range are often restricted to nonnegative or whole numbers.
Ranjan writes and edits short articles for a local newspaper. It takes him about an hour to write an article and about a half-hour to edit an article. If Ranjan works up to 8 hours a day, how many articles can he write and edit in one day?
x + ½y ≤ 8
-x x
2( )
½y ≤ -x + 8
y ≤ -2x + 16
Test Point: (0, 0)
0 ≤ -2(0) + 16
Any point in the shaded area is a solution.
0 ≤ 16
True
Sample Answer: (2, 4)
He can write 2 and edit 4 articles.

11. Check Your Progress Choose the best answer for the following.
You offer to go to the local deli and pick up sandwiches for lunch. You have $30 to spend. Chicken sandwiches cost $3.00 each and tuna sandwiches are $1.50 each. How many sandwiches can you purchase for $30?
11 chicken sandwiches, 1 tuna sandwiches
12 chicken sandwiches, 3 tuna sandwiches
3 chicken sandwiches, 15 tuna sandwiches
5 chicken sandwiches, 9 tuna sandwiches
3x y ≤ 30
3(11) (1) ≤ 30
34.5 ≤ 30
False
3(12) (3) ≤ 30
40.5 ≤ 30
False
3(3) (15) ≤ 30
31.5 ≤ 30
False
3(5) (9) ≤ 30
28.5 ≤ 30
True