1. Math in Our World
Section 7.4
Linear Inequalities

2. Learning Objectives Graph linear inequalities in two variables.
Graph a system of linear inequalities.
Model a real-world situation with a system of linear inequalities.

3. Matrices We know that an equation of the form ax + by = c
is called a linear equation in two variables, and that its graph is a straight line. When the equal sign is replaced by >, <, ≥, or ≤, the equation becomes a linear inequality in two variables.
2x + y > 6 3x – 8y ≤ 20 x – y ≥ 3

4. Linear Inequalities in Two Variables
Procedure for Graphing a Linear Inequality
Step 1 Replace the inequality symbol with an equal sign and graph the line.
(a) Use a dashed line if the inequality is either > or < (a strict inequality) to indicate that the points on the line are not included in the solution.
(b) Use a solid line if the inequality is either ≥ or ≤ to indicate that the points on the line are included in the solution.

5. Linear Inequalities in Two Variables
Procedure for Graphing a Linear Inequality
Step 2 Pick any point not on the line and substitute its coordinates into the inequality to see if the resulting statement is true or false.
Step 3 The line divides the plane into two half planes.
(a) If the test point makes the inequality true, shade the half plane containing that point.
(b) If the test point makes the inequality false, shade the half plane not containing that point.
The shaded region is the solution set of the inequality.

6. EXAMPLE 1 Graphing a Linear Inequality
Graph x – y ≥ 6.
SOLUTION
Step 1 Graph the line x – y = 6. It will be easy to find the intercepts, and we’ll choose one other point to be safe.
Since the sign is ≥, equality is included. That means the points on the line satisfy the inequality, so we draw a solid line through the three points.

7. EXAMPLE 1 Graphing a Linear Inequality
SOLUTION
Step 2 Pick a test point not on the line, substitute into the equation, and see if a true or false statement results.
For (0, 0):
Step 3 The point (0, 0) is above the line, and is not in the solution set for the inequality, so we shade the half-plane below the line.

8. EXAMPLE 2 Graphing a Linear Inequality
Graph 2x – 4y < 12.
SOLUTION
Step 1 Graph the line 2x – 4y = 12.
Since the sign is <, equality is not included. That means the points on the line do not satisfy the inequality, so we draw a dashed line through the three points.

9. EXAMPLE 2 Graphing a Linear Inequality
SOLUTION
Step 2 Pick a test point not on the line, substitute into the equation, and see if a true or false statement results.
For (0, 0): - -
Step 3 The point (0, 0) is below the line, and is in the solution set for the inequality, so we shade the half-plane below the line.

10. Systems of Linear Inequalities
Systems of linear equations were solved graphically by graphing each line and finding the point where the lines intersect.
Systems of linear inequalities can be solved graphically by graphing each inequality and finding the intersection of the shaded regions.

11. EXAMPLE 3 Solving a System of Linear Inequalities
Solve the system graphically:
SOLUTION
Begin by graphing the first inequality, x + 3y ≥ 6, as in Example 1. The intercepts are (6, 0) and (0, 2), and we draw a solid line through those points since equality is included.
Pick (0, 0) as a test point:
0 + 3(0) ≥ 6 is false, so we shade the half-plane above the line, which does not contain (0, 0).

12. EXAMPLE 3 Solving a System of Linear Inequalities
Solve the system graphically:
SOLUTION
Next we add the second inequality to the graph, starting with the points (5, 0) and (4, 2). The line is dashed because equality is not included in 2x – y < 10.
Pick (0, 0) as a test point:
2(0) – 0 < 10 is a true statement, so we shade the region to the left of the dashed line.

13. EXAMPLE 3 Solving a System of Linear Inequalities
Solve the system graphically:
SOLUTION
The solution to the system is the intersection of the two shaded regions.
The points on the solid line are included, but those on the dashed line are not. (Note the open circle where the lines intersect.)

14. EXAMPLE 4 Solving a System of Linear Inequalities
Solve the system graphically: 2
SOLUTION
The inequality x > – 3 represents the set of all points whose x coordinates are larger than – 3. This is all points to the right of the dashed vertical line x = – 3, shaded in blue.
The inequality y ≤ 2 is the set of all points whose y coordinate is less than or equal to 2. This is all points on or under the solid horizontal line y = 2, shaded in pink.

15. EXAMPLE 4 Solving a System of Linear Inequalities
Solve the system graphically: 2
SOLUTION
The solution set to the inequality is the intersection of those regions, shaded in purple.

16. EXAMPLE 5 Applying Systems of Linear Inequalities to Purchasing
You are dispatched to buy cups and plates for a homecoming tailgate event. The plates you choose are $0.40 each, and the cups are $0.25 each. You were told that any more than $30 is coming out of your own pocket (unacceptable, obviously), and that you have to get at least 40 plates. Draw a graph that represents acceptable numbers of plates and cups.

17. EXAMPLE 5 Applying Systems of Linear Inequalities to Purchasing
SOLUTION
The two quantities that can vary are the number of plates purchased, and the number of cups. We will let
x = the number of plates
y = the number of cups
The cost of plates will be 0.40x (40 cents times number of plates). The cost of cups will be 0.25y (25 cents times the number of cups). The total cost is the sum, and it has to be $30 or less.
0.40x y ≤ 30
We need at least 40 plates, so x ≥ 40. Finally, the number of cups certainly can’t be negative, so y ≥ 0.

18. EXAMPLE 5 Applying Systems of Linear Inequalities to Purchasing
SOLUTION
Now we have a system:
0.40x y ≤ 30
x ≥ 40
y ≥ 0
To graph the first inequality, find the intercepts and choose (0, 0) as test point:

19. EXAMPLE 5 Applying Systems of Linear Inequalities to Purchasing
SOLUTION
Now we have a system:
0.40x y ≤ 30
x ≥ 40
y > 0
The inequality x ≥ 40 represents all points on or to the right of the vertical line x = 40.

20. EXAMPLE 5 Applying Systems of Linear Inequalities to Purchasing
SOLUTION
Now we have a system:
0.40x y ≤ 30
x ≥ 40
y > 0
The inequality y ≥ 0 is all points on or above the horizontal line y = 0, which is the x axis.

21. EXAMPLE 5 Applying Systems of Linear Inequalities to Purchasing
SOLUTION
Now we have a system:
0.40x y ≤ 30
x ≥ 40
y > 0
The intersection of the shaded regions is the triangular region to the left. Any combination of plates (x coordinate) and cups (y coordinate) from points in that region is an acceptable solution.
For example, the point (40, 40) is in the region, so 40 cups and 40 plates will do.