Lesson Objectives

- Demonstrate an understanding of how to graph a Linear Inequality in Two Variables
- Learn how to graph a System of Linear Inequalities in Two Variables

## How to Graph a System of Linear Inequalities in Two Variables

In a previous lesson, we learned how to graph a linear inequality in two variables. We learned that we would first solve our inequality for y and graph our boundary line by replacing our inequality symbol with an equality symbol. We would graph our boundary line as a solid line for a non-strict inequality and a dashed line for a strict inequality. Once this is done, we can shade above the line for a greater than or below the line for a less than. When we solve a system of linear inequalities, we are looking for the section of the coordinate plane that represents the overlap from the two graphs. This means the area that will be a solution to both inequalities. Let’s take a look at a few examples.

Example 1: Graph each system of linear inequalities.

1) x + y ≥ -3

2) x - y ≥ 1

Let's begin by solving each inequality for y:

1) y ≥ -x - 3

2) y ≤ x - 1

Let's graph our first inequality. We first graph the solid boundary line of:

y = -x - 3

Then we will shade above the line: Let's now graph our second inequality. We first graph the solid boundary line of:

y = x - 1

Then we will shade below the line: Let's overlay our two graphs and look for the overlap. In other words the portion of the coordinate plane which satisfies both inequalities: Now that we have found our overlap or section of the coordinate plane that satisfies both linear inequalities, we can draw a graph with just this portion shaded. This will be the solution for our system: Let's look at another example.

Example 2: Graph each system of linear inequalities.

1) 5x + 3y < 21

2) 8x - 3y > 18

Let's begin by solving each inequality for y:

1) y < -5/3 x + 7

2) y < 8/3 x - 6

Let's graph our first inequality. We first graph the dashed boundary line of:

y = -5/3 x + 7

Then we will shade below the line: Let's now graph our second inequality. We first graph the dashed boundary line of:

y = 8/3 x - 6

Then we will shade below the line: Let's overlay our two graphs and look for the overlap. In other words the portion of the coordinate plane which satisfies both inequalities: Now that we have found our overlap or section of the coordinate plane that satisfies both linear inequalities, we can draw a graph with just this portion shaded. This will be the solution for our system:

Example 3: Graph each system of linear inequalities.

1) 4x - y < -2

2) 8x - 2y > 4

Let's solve each for y:

1) y > 4x + 2

2) y < 4x - 2

We can see that we will have parallel boundary lines. The slope in each case will be 4. Let's graph each dashed boundary line. We will shade above for inequality 1 and below for inequality 2: We can see from our graph that there is no overlap. This means there is no area of the coordinate plane that satisfies both inequalities of our system. This is a situation where we have a system of linear inequalities with no solution. For our answer, we can simply write "no solution".

Example 1: Graph each system of linear inequalities.

1) x + y ≥ -3

2) x - y ≥ 1

Let's begin by solving each inequality for y:

1) y ≥ -x - 3

2) y ≤ x - 1

Let's graph our first inequality. We first graph the solid boundary line of:

y = -x - 3

Then we will shade above the line: Let's now graph our second inequality. We first graph the solid boundary line of:

y = x - 1

Then we will shade below the line: Let's overlay our two graphs and look for the overlap. In other words the portion of the coordinate plane which satisfies both inequalities: Now that we have found our overlap or section of the coordinate plane that satisfies both linear inequalities, we can draw a graph with just this portion shaded. This will be the solution for our system: Let's look at another example.

Example 2: Graph each system of linear inequalities.

1) 5x + 3y < 21

2) 8x - 3y > 18

Let's begin by solving each inequality for y:

1) y < -5/3 x + 7

2) y < 8/3 x - 6

Let's graph our first inequality. We first graph the dashed boundary line of:

y = -5/3 x + 7

Then we will shade below the line: Let's now graph our second inequality. We first graph the dashed boundary line of:

y = 8/3 x - 6

Then we will shade below the line: Let's overlay our two graphs and look for the overlap. In other words the portion of the coordinate plane which satisfies both inequalities: Now that we have found our overlap or section of the coordinate plane that satisfies both linear inequalities, we can draw a graph with just this portion shaded. This will be the solution for our system:

### Solving Systems of Linear Inequalities with No Solution

We previously looked at systems of linear equations with no solution. This occurs when we have parallel lines. Since parallel lines will never intersect, there will never be a solution for such a system. A system of linear inequalities that contains parallel boundary lines may have no solution. It will depend on the inequality symbols present in the system. Let's take a look at an example.Example 3: Graph each system of linear inequalities.

1) 4x - y < -2

2) 8x - 2y > 4

Let's solve each for y:

1) y > 4x + 2

2) y < 4x - 2

We can see that we will have parallel boundary lines. The slope in each case will be 4. Let's graph each dashed boundary line. We will shade above for inequality 1 and below for inequality 2: We can see from our graph that there is no overlap. This means there is no area of the coordinate plane that satisfies both inequalities of our system. This is a situation where we have a system of linear inequalities with no solution. For our answer, we can simply write "no solution".

#### Skills Check:

Example #1

Determine which system matches the graph.

Please choose the best answer.

A

$$5x + 3y > 4$$ $$x - 2y ≤ 7$$

B

$$2x - 5y > 4$$ $$-x - 3y ≤ -4$$

C

$$-x + y > 7$$ $$8x + 4y ≤ 3$$

D

$$5x - 3y > 6$$ $$x - 3y ≤ -6$$

E

$$4x - 2y > 5$$ $$x - 2y ≤ -6$$

Example #2

Determine which system matches the graph.

Please choose the best answer.

A

$$5x + 4y ≤ 1$$ $$-x - 4y < 10$$

B

$$2x + 3y ≤ -9$$ $$4x - 3y < -9$$

C

$$x + 2y ≤ -5$$ $$x + 5y < 7$$

D

$$4x + 7y ≤ 10$$ $$-x - 7y < 12$$

E

$$3x + 8y ≤ 11$$ $$-x - 3y < 14$$

Example #3

Determine which system matches the graph.

Please choose the best answer.

A

$$x + 7y ≤ 11$$ $$x + 3y ≤ 10$$

B

$$5x - 4y ≤ 10$$ $$2x - 3y ≤ 6$$

C

$$x + 10y ≤ 14$$ $$-5x + 8y ≤ 11$$

D

$$7x + 3y ≤ 11$$ $$-2x - 3y ≤ 15$$

E

$$2x + y ≤ -3$$ $$3x - y ≤ -2$$

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