1. Objective
solve systems of linear inequalities in two variables

2. Section 6.7 “Graph Linear Inequalities”
Remember
This???
Linear Inequalities-
the result of replacing the = sign
in a linear equation with an inequality sign.
2x + 3y > 4
y ≥ 4x - 3
y ≤ ½x + 3
7y < 8x - 16

3. Graphing Linear Inequalities
Remember
This???
Graphing Linear Inequalities
Graphing Boundary Lines:
Use a dashed line for < or >.
Use a solid line for ≤ or ≥.

4. Graph the inequality y > 4x - 3.
Graph an Inequality
Remember
This???
Graph the inequality y > 4x - 3.
STEP 2
STEP 3
STEP 1
Graph the equation
Test (0,0) in the original inequality.
Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality.

5. Graph the inequality x + 3y ≥ -1.
Graph an Inequality
Remember
This???
Graph the inequality x + 3y ≥ -1.
STEP 2
STEP 3
STEP 1
Shade the half-plane that contains the point (1,0), because (1,0) is a solution to the inequality.
Graph the equation
Test (1,0) in the original inequality.

6. Graph the inequality y ≥ -3.
Graph an Inequality
Remember
This???
Graph the inequality y ≥ -3.
STEP 2
STEP 3
STEP 1
Shade the half-plane that contains the point (2,0), because (2,0) is a solution to the inequality.
Graph the equation
Test (2,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

7. Graph the inequality x ≤ -1.
Graph an Inequality
Remember
This???
Graph the inequality x ≤ -1.
STEP 2
STEP 3
STEP 1
Shade the half-plane that does not contain the point (3,0), because (3,0) is not a solution to the inequality.
Graph the equation
Test (3,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

8. Section 7.6 “Solve Systems of Linear Inequalities”
SYSTEM OF INEQUALITIES-
consists of two or more linear inequalities in the same variables.
x – y > 7
Inequality 1
2x + y < 8
Inequality 2
A solution to a system of inequalities is an ordered pair (a point) that is a solution to both linear inequalities.

9. Solving a System of Inequalities by Graphing
(1) Graph both inequalities in the same plane.
(2) Find the intersection of the two half-planes. The graph of the system is this intersection.
(3) Check a coordinate by substituting into EACH inequality of the system, to see if the point is a solution for both inequalities.

10. Graph a System of Inequalities
y > -x – 2
Inequality 1
y ≤ 3x + 6
Inequality 2
Graph both inequalities in the same coordinate plane. The graph of the system is the intersection of the two half-planes, which is shown as the darker shade of blue.
(0,1)
1 > 0 – 2 ?
1 ≤ 0 + 6 ?
1 > – 2
1 ≤ 6

11. Graph a System of Inequalities
y < x – 4
Inequality 1
y ≥ -x + 3
Inequality 2
Graph both inequalities in the same coordinate plane. The graph of the system is the intersection of the two half-planes, which is shown as the darker shade of blue.
(5,0)
0 < 5 – 4 ?
0 ≥ ?
0 < 1
0 ≥ -2

12. Graph a System of THREE Inequalities
Inequality 1:
Graph all three inequalities in the same coordinate plane. The graph of the system is the triangular region, which is shown as the darker shade of blue.
x > -2
Inequality 2:
x + 2y ≤ 4
Inequality 3:
x + 2y ≤ 4 ?
y ≥ -1 ?
x > -2 ?
Check
(0,0)
0 ≥ -1
0 > -2
0 + 0 ≤ 4

13. Graph a System of THREE Inequalities
y ≥ -x + 2
y > -x
Inequality 1:
Inequality 1:
y < 4
y ≥ x – 4
Inequality 2:
Inequality 2:
x < 3
y < 5
Inequality 3:
Inequality 3:

14. Write a System of Linear Inequalities
Write a system of inequalities for the shaded region.
INEQUALITY 1: One boundary line for the shaded region is y = 3. Because the shaded region is above the solid line, the inequality is y ≥ 3.
INEQUALITY 2: Another boundary line for the shaded region has a slope of 2 and a y-intercept of 1. So, its equation is y = 2x + 1. Because the shaded region is above the dashed line, the inequality is y > 2x + 1.
y ≥ 3
y > 2x + 1
Inequality 1
Inequality 2

15. NJASK7 Prep
Homework
Text p. 469, #3-8 all, #10-38 evens