# Algebra

## Systems of Linear Inequalities in Two Variables


### Learning Objectives

After completing this section, you should be able to:
1. Demonstrate whether an ordered pair is a solution to a system of linear inequalities.
2. Solve systems of linear inequalities using graphical methods.
3. Graph systems of linear inequalities.
4. Interpret and solve applications of linear inequalities.
In this section, we will learn how to solve systems of linear inequalities in two variables. In Systems of Linear Equations in Two Variables, we learned how to solve for systems of linear equations in two variables and found a solution that would work in both equations. We can solve systems of inequalities by graphing each inequality (as discussed in Graphing Linear Equations and Inequalities) and putting these on the same coordinate system. The double-shaded part will be our solution to the system. There are many real-life examples for solving systems of linear inequalities.
Consider Ming who has two jobs to help her pay for college. She works at a local coffee shop for $7.50 per hour and at a research lab on campus for $12 per hour. Due to her busy class schedule, she cannot work more than 15 hours per week. If she needs to make at least $150 per week, can she work seven hours at the coffee shop and eight hours in the lab?

### Determining If an Ordered Pair Is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is similar to the definition of a system of linear equations. A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.
To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs  that make both inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the -coordinate system that includes all the points whose ordered pairs make the inequalities true.
To determine if an ordered pair is a solution to a system of two inequalities, substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

### Solving Systems of Linear Inequalities Using Graphical Methods

The solution to a single linear inequality was the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. We will review graphs of linear inequalities and solve the linear inequality from its graph.

### Graphing Systems of Linear Inequalities

We learned that the solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region by graphing, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.
Step 1: Graph the first inequality.
Graph the boundary line.
Shade in the side of the boundary line where the inequality is true.
Step 2: On the same grid, graph the second inequality.
Graph the boundary line.
Shade in the side of that boundary line where the inequality is true.
Step 3: The solution is the region where the shading overlaps.
Step 4: Check by choosing a test point.
Systems of linear inequalities where the boundary lines are parallel might have no solution. We will see this in the next example.
Some systems of linear inequalities where the boundary lines are parallel will have a solution. We will see this in the next example.

### Interpreting and Solving Applications of Linear Inequalities

When solving applications of systems of inequalities, first translate each condition into an inequality. Then graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so add inequalities to the system as additional requirements.

### Check Your Understanding


### Key Terms

1. system of linear inequalities

### Key Concepts

1. To solve a system of linear inequalities means to find the area(s) where the points in that area make all the linear inequalities true.
2. Systems of linear inequalities can be solved by graphing the linear equations associated with the inequalities, then 'testing' points to see whether the values of the point make the equation true or not.

### Videos

1. Solving Systems of Linear Inequalities by Graphing
2. Systems of Linear Inequalities