System of Linear Inequalities and Equations

Introduction:

Systems of Linear Inequalities:

A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this,

y < x + 2…………..Inequality 1

y ≥ 2x − 1…………Inequality 2

Solution of a System of Linear Inequalities:

A solution of a system of linear inequalities in two variables is an ordered pair that is a solution of each inequality in the system. The solution is the region where the two shaded regions intersect.

Graph a System of Inequalities

Example 1:

What is the solution to the following system of linear inequalities?

x + y < 4

−x + y ≤ 3

Solution:

Step 1: Graph each inequality.

Step 2: Find the intersection of the half-planes. One solution is (1, −3).

Example 2:

What is the solution to the following system of linear inequalities?

y ≤ 3

y > 5

Solution:

Step 1: Graph each inequality.

Step 2: Since the slopes of the boundary lines are equal and their y-intercepts are different, they are parallel and do not intersect.

Write a System of Inequalities from a Graph

Example 3:

What system of inequalities is shown in the following graph?

Solution:

Determine the equation of each line using the slope and y-intercept.

The slope of the blue boundary line is 1, and it has a y-intercept of 3.

The slope of the red boundary line is 0, and it has a y-intercept of –3.

The solutions to the system are below the solid blue line, so one inequality is y <= x + 3.

The solutions to the system are below the dashed red line, so one inequality is y < -3.

The graph shows the system of inequalities, y <= x + 3 and y < -3.

Use a System of Inequalities

Example 4:

Mike has at most 8 hours to spend at the shopping mall and at the beach. He wants to spend at least 2 hours at the shopping mall and more than 4 hours at the beach. Write and graph a system that represents the situation. How much time could he spend at each location?

Solution:

Formulate:

Let x = the number of hours at the mall.

Let y = the number of hours at the beach.

Write a system of inequalities.

x + y ≤ 8……………Mikes wants to spend at most 8 hours at the shopping mall and at the beach.

x ≥ 2………………….. Mike wants to spend at least 2 hours at the shopping mall.

y > 4…………………. Mike wants to spend more than 4 hours at the beach.

Compute:

Graph the system of inequality.

Interpret:

Any point in the shaded region is a solution to inequality. One ordered pair in the solution region is (2.5, 5). Therefore, Mike could spend 2.5 hours at the shopping mall and 5 hours at the beach.

Exercise

1. Two or more linear inequalities in two variables form a(n) ___________________________.
2. A __________________ of a system of linear inequalities in two variables is an ordered pair that is a solution of each inequality in the system.
3. What system of inequalities is shown in the following graph?

4. What is the solution to the following system of linear inequalities?

y < 2x

y > –3

5. What is the solution to the following system of linear inequalities?

y >= –2x + 1

y > x + 2

6. What is the solution to the following system of linear inequalities

y < 2x + 1

y <= –x – 4

7. What system of inequalities is shown in the following graph?

8. What system of inequalities is shown in the following graph?

9. How is a system of two linear inequalities in two variables similar to a system of two linear equations in two variables? How is it different?

10. A person is planning a weekly workout schedule of cardio and yoga. He has at most 12 hours per week to work out. The amount of time he wants to spend on cardio and yoga is shown.

Write a system of linear inequalities to represent this situation.

Concept Map: