3-3 Graphing and Solving Systems of Linear Inequalities Objective: Students will be able to graph a system of inequalities to find the solutions of the.

3-33-3Graphing and Solving Graphing and Solving

Systems of Linear Systems of Linear InequalitiesInequalities

Objective: Students will be able to graph a system of inequalities to find the solutions

of the system.

The following is a system of linear inequalities.

6

2 4

x y

x y

A solution of a system of linear inequalities is an ordered pair that is a solution of each

inequality in the system.

The graph of a system of linear inequalities is the graph of all solutions of the system.

Graphing a System of Graphing a System of Linear InequalitiesLinear Inequalities

1. Graph the boundary line that corresponds to the inequality. Use a

dashed lined for an inequality with < or > and a solid line for or .

2. Lightly shade in the half-plane that is the graph of the inequality. Colored pencils may help you distinguish the different half-planes.

Example 1: Graph the system

3 1

2

y x

y x

Step 1: Graph the boundary line for each equation.

3 1 Solid line

2 Dashed line

y x

y x

y > -3x + 1

y < x + 2

Determine the half planes that make the inequalities true.

Choose a test point (0, 0)

0 3 0 1

0 1 is a true statement shade the area to the right.

0 0 2

0 2 is a true statement shade the area to the right.

Example 2:

Graph the system 0

0

4 3 0

x

y

x y

10

8

6

4

2

5

(1, 1)

4x+3y=24

y=0

x=0

Test point (1, 1)

4 1 3 1 24

4 3 24

7 24 The statement is true shade the region containing the test point.

Example 3

Writing and using a system of Inequalities

Heart Rate: A person’s theoretical maximum heart rate is 220 – x where x is the person’s age in years (20 x 65). When a person exercises, it is recommended that the person strive for a heart rate that is at least 70% and at most 85% of the maximum.

a. You are making a poster for the health class. Write and graph a system of linear inequalities that describes the information given above.

b. A 40- year old person has a heart rate of 150 (heart beat per minute) when exercising. Is the person’s heart rate in the target zone?

Solution:

Let y represent the person’s heart rate. From the given information, you can write the following inequalities

20 Person's age must be at least 20 years old

65 Person's age can be at most 65.

y 0.7 200-x Target rate is at least 70% of maximum.

y 0.85 200-x Target rate is at most 85% o

x

x

f maximum.

Solve:

y 0.7 220-20 y 0.7 220 65

0.7 200 0.7(155)

140 108.5

y y

y y

This determines the lower limit of the target zone.

0.85 220 20 0.85 220 65

0.85 200 0.85 155

170 131.75

y y

y y

y y

This determines the upper limit of the target zone.

b. From the graph you can see that the target zone for a 40 year-old person is between 126 and 153. A 40 year-old person who has a heart rate of 150 is within the target zone.

Homework page 159, 12 – 46 even, 51-54, 67 – 77 odd

Turn to page 156 in your book. Investigating Graphs of Systems of Inequalities. Work in groups. Stay on task. You will have 10 minutes to complete the task.

a) Region 1

b) Region 3

c) Region 4

d) Region 2