Lesson Objectives
• Demonstrate the ability to graph a linear inequality in two variables
• Learn how to graph a system of linear inequalities in two variables
• Learn how to identify a system of linear inequalities with "no solution"

## How to Graph a System of Linear Inequalities

We previously learned how to graph a linear inequality in two variables. Let's review this procedure and then move into solving systems of linear inequalities.

### Graphing a Linear Inequality in Two Variables

• Solve the inequality for y
• Graph the boundary line
• Replace the inequality symbol with an equality symbol
• Graph the resulting equation
• The boundary line is solid for a non-strict inequality
• The boundary line is dashed for a strict inequality
• When the inequality is solved for y:
• We shade above the boundary line for a greater than
• We shade below the boundary line for a less than

### Graphing a System of Linear Inequalities

• Graph each inequality of the system
• Shade the solution region for the system
• The solution region for the system is the area of the coordinate plane that satisfies both inequalities
• We can think of the solution region as the overlap of the graphs
When we graph a system of linear inequalities in two variables, we graph each inequality and look for the overlap of the graphs. This overlap is the solution region for the system since it is the area of the coordinate plane that satisfies both inequalities of the system. Let's look at a few examples.
Example 1: Graph each. $$2x + y ≥ -2$$ $$x - 2y ≥ -6$$ Let's solve each inequality for y: $$y ≥ -2x - 2$$ $$y ≤ \frac{1}{2}x + 3$$ Let's graph our top inequality first: $$y ≥ -2x - 2$$ Now let's graph our bottom inequality: $$y ≤ \frac{1}{2}x + 3$$ The solution for the system is the overlap of the two graphs: Example 2: Graph each. $$2x - y ≤ 5$$ $$4x + y > 7$$ Let's solve each inequality for y: $$y ≥ 2x - 5$$ $$y > -4x + 7$$ Let's graph our top inequality first: $$y ≥ 2x - 5$$ Now let's graph our bottom inequality: $$y > -4x + 7$$ The solution for the system is the overlap of the two graphs:

### Systems of Linear Inequalities with No Solution

In some cases, we will see a system of linear inequalities with no solution. This will occur when the boundary lines are parallel and there is no overlap between the two graphs. Let's take a look at an example.
Example 3: Graph each. $$6x + y < -5$$ $$-18x - 3y < 9$$ Let's solve each inequality for y: $$y < -6x - 5$$ $$y > -6x - 3$$ Before we graph anything, we can see that our boundary lines are going to be parallel. The slopes are the same and they each have a different y-intercept. Although we may still have a solution when parallel boundary lines occur, we should be prepared for a scenario in which there is "no solution".
Let's graph our top inequality first: $$y < -6x - 5$$ Now let's graph our bottom inequality: $$y > -6x - 3$$ When we graph the two inequalities on the same coordinate plane, we see there is no overlap. This means there is no section of the coordinate plane that satisfies both inequalities and, therefore, there is no solution for the system. No Solution ∅

#### Skills Check:

Example #1

Determine which system matches the graph.

A
$$2x + 5y > 7$$ $$x - 3y ≤ -2$$
B
$$x - 2y > 4$$ $$2x - y ≤ -1$$
C
$$2x + 7y < -2$$ $$4x + 5y ≥ -3$$
D
$$3x + y < 1$$ $$x - y ≥ -1$$
E
$$3x + 5y > 2$$ $$3x + 4y ≤ 7$$

Example #2

Determine which system matches the graph.

A
$$x - 3y < -6$$ $$2x + 3y < 6$$
B
$$x + 3y < 6$$ $$-x - 3y > 5$$
C
$$2x - 7y > -2$$ $$4x + 5y > 8$$
D
$$3x + 9y < 1$$ $$x - 2y < 4$$
E
$$5x + 3y < -6$$ $$x + 3y < 6$$

Example #3

Determine which system matches the graph.

A
$$5x - 3y > 7$$ $$x + 4y ≤ 2$$
B
$$2x + 5y > 7$$ $$13x - 9y ≤ 7$$
C
$$x + y > -2$$ $$4x - y ≤ -3$$
D
$$2x - 5y < -9$$ $$x - 3y ≥ 8$$
E
$$4x - 7y ≤ -1$$ $$x + 13y > 7$$