1. 6. 5 Graphing Linear Inequalities in Two Variables 7
6.5 Graphing Linear Inequalities in Two Variables 7.6 Graphing Linear Inequalities System
Objectives
1. Graph linear inequalities.
2. Graph systems of linear inequalities.

2. Linear Inequalities
A linear inequality in two variables is an inequality that can be written in the form
Ax + By < C,
where A, B, and C are real numbers and A and B are not both zero. The symbol < may be replaced with , >, or .
The solution set of an inequality is the set of all ordered pairs that make it true. The graph of an inequality represents its solution set.

3. To Graph a Linear Inequality
Step
(1) Solve for y, convert the inequalities to Slope-Intercept Form. If one variable is missing, solve it and go to step (2).
(2) Graph the related equation.
** If the inequality symbol is < or >, draw the line dashed.
** If the inequality symbol is  or , draw the line solid.
(3) If y < or y  , shade the region BELOW the line
If y > or y  , shade the region ABOVE the line
If x < or x  , shade the region LEFT to the line
If x > or x  , shade the region RIGHT to the line

4. Example Graph y > x  4. Already in S-I form.
The related equation is y = x  4. Use a dashed line because the inequality symbol is >. This indicates that the line itself is not in the solution set.
(3) Determine which half-plane satisfies the inequality.
y > x  4
“>” shade above

5. Example Graph: 4x + 2y  8 1. Convert to S-I form: y  -2x + 4
2. Graph the related equation.
3. Determine which half-plane satisfies the inequality.
“y ” shade below

6. Example Graph 2x – 4 > 0 . 1. One variable is missing. Solve it:
2. Graph the related equation
x = 2
3. Determine which half-plane satisfies the inequality.
“x >” shade the right.

7. Example Graph 8 - 3y  2 . 1. One variable is missing. Solve it: y  2
2. Graph the related equation
y = 2
3. Determine the region to be shaded.
“y ” shaded below

8. 7.6 Systems of Linear Inequalities
Graph the solution set of the system.
First, we graph x + y  3 using a solid line.
y  - x + 3
“above”
Next, we graph x  y > 1 using a dashed line.
y < x – 1
“below”
The solution set of the system of equations is the region shaded both red and green, including part of the line x + y = 3.

9. Example
Graph the following system of inequalities and find the coordinates of any vertices formed:
We graph the related equations using solid lines. We shade the region common to all three solution sets.

10. Example continued
The system of equations from inequalities (1) and (3):
y + 2 = 0
x + y = 0
The vertex is (2, 2).
The system of equations from inequalities (2) and (3):
x + y = 2
The vertex is (1, 1).
To find the vertices, we solve three systems of equations. The system of equations from inequalities (1) and (2):
y + 2 = 0
x + y = 2
The vertex is (4, 2).

11. Summary
To graph a two-variable linear inequality, graph the related equation first (variable y must be solved) with appropriate boundary line:
“<” and “>” use dash line
“≤” and “≥” use solid line
“y < …” and “y ≤ …” shade the region below
“y > …” and “y ≥ …” shade the region above
“x < …” and “x ≤ …” shade the region left
“x > …” and “x ≥ …” shade the region right
When graphing the linear inequality system, follow the step 1 ~ 3 and choose the region shaded most.

12. Assignment
6.5 P 363 #’s (even), (even), (even)
7.6 P 435 #’s