Intermediate Algebra
Tutorial 17: Graphing Linear Inequalities

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Learning Objectives

After completing this tutorial, you should be able to:

Graph linear inequalities in two variables.
Graph the union of two linear inequalities in two variables.
Graph the intersection of two linear inequalities in two variables.

Introduction

In this tutorial we will be looking at linear inequalities in two variables. It will start out exactly the same as graphing linear equations and then we get to color in the region of the coordinate system that correlates with the inequality. Some of these problems may get a little long, especially the ones that involve two inequalities. Don't let that discourage you, you can do it. Hang in there, a lot of the steps are concepts from the past, things you should already have seen and done before. I will put in links to the material that you need to remember from the past, in case you need a review. A lot of times math works that way, use what you know to learn the new concept. Let's see what you can do with these inequalities.

Tutorial

Linear Inequalities in Two Variables

A linear inequality in two variables is any expression that can be put in the form

where a, b, and c are constants

Note that the above definition can be applied to any of the following four inequalities: <, >, <, or >.

The solution set and graph for a linear inequality is a region of the rectangular coordinate system. Recall that the graph of a linear equation is a straight line. The inequality sign extends this to being on one side of the line or the other on the graph.

Graphing a Linear Inequality

Step 1: Graph the boundary line.

Essentially you graph the boundary line the same as if the problem was a linear equation. Pretend that there is an equal sign and use an appropriate method to graph the line. Unless the directions to a problem indicate otherwise, you can use any method to graph it. The ways covered in our past tutorials are plotting any three points as shown in Tutorial 12: Graphing Equations, using the x and y intercepts as shown in Tutorial 14: Graphing Linear Equations, or using the slope and y-intercept as shown in Tutorial 16: Equations of Lines.

When you draw the boundary line, you need to have a way to indicate if the line is included or not in the final answer.

Solid boundary line: < or >
If the problem includes where it is equal, then you will have a solid boundary line. In other words, if you have < or > , you will have a solid line for your boundary line.

This shows the boundary line for x + y < 6:
(note that this does not show the inequality part)

solid line

Dashed boundary line: < or >
If the problem does not include where it is equal, then you will use a dashed boundary line. In other words, if you have < or >, you will have a dashed line for your boundary line.

This shows the boundary line for x + y < 6:
(note that this does not show the inequality part)

dash line

In either case, you still graph the line the same. You just have to decide if you are needing a solid line or a dashed line.

The boundary line separates the rectangular coordinate system into two parts. One of those parts will make the inequality true and be it’s solution.

Step 2: Plug in a test point that is not on the boundary line.

Pick a test point on either side of the boundary line and plug it into the original problem. This will help determine which side of the boundary line is the solution.

Step 3: Shade in the answer to the inequality.

If you get a true statement when you plug in the test point in step 2, then you have found a solution. Shade the region that the test point is in.

If you get a false statement when you plug in the test point in step 2, then you don’t have a solution. Shade in the region that is on the other side of the test point.

It doesn’t matter what you use for the test point as long as it is not on the
boundary line. You want to keep it as simple as possible.

Example 1: Graph the inequality

Step 1: Graph the boundary line.

I’m going to use the intercepts to help me graph the boundary line. Again, you can use any method that you want, unless the directions say otherwise.

When I’m working with only the boundary line, I will put an equal sign between the two sides to emphasize that we are working on the boundary line. That doesn’t mean that I changed the problem. When we put it all together in the end, I will put the inequality back in.

What value is y on the x-intercept?
If you said 0, you are correct.
If you need a review on x-intercepts, go to Tutorial 14: Graphing Linear Equations.

*Replace y with 0
*x-intercept

x-intercept is (2, 0)

What is the value of x on the y-intercept?
If you said 0, you are correct.
If you need a review on y-intercepts, go to Tutorial 14: Graphing Linear Equations.

*Replace x with 0
*y-intercept

y-intercept is (0, 2).

Plug in 1 for x to get a third solution:

*Replace x with 1
*Inverse of add 1 is sub. 1

(1, 1) is another solution on the boundary line.

Solutions:

x	y	*(x, y)*
2	0	(2, 0)
0	2	(0, 2)
1	1	(1, 1)

Since the original problem has a >, this means it DOES NOT include the boundary line.

So are we going to draw a solid or a dashed line for this problem?
It looks like it will have to be a dashed line.

Putting it all together, we get the following boundary line for this problem:

example 1e

Step 2: Plug in a test point that is not on the boundary line.

Note how the boundary line separates it into two parts.

An easy test point would be (0, 0). Note that it is a point that is not on the boundary line. In fact, it is located below the boundary line.

Let’s put (0, 0) into the original problem and see what happens:

*Replacing x and y with 0
*False statement

Step 3: Shade in the answer to the inequality.

Since our test point (0, 0) made our inequality FALSE, this means it is not a solution.

Since it has to be on one side or the other of the boundary line, and it is not below it, then our solution would lie above the boundary line. This means we will shade in the part that is above it.

Note that the gray lines indicate where you would shade your final answer.

Example 2: Graph the inequality

Step 1: Graph the boundary line.

I’m going to use the intercepts to help me graph the boundary line. Again, you can use any method that you want, unless the directions say otherwise.

What value is y on the x-intercept?
If you said 0, you are correct.
If you need a review on x-intercepts, go to Tutorial 14: Graphing Linear Equations.

*Replace y with 0

*Inverse of mult. by 2 is div. by 2

*x-intercept

x-intercept is (3, 0)

What is the value of x on the y-intercept?
If you said 0, you are correct.
If you need a review on y-intercepts, go to Tutorial 14: Graphing Linear Equations.

*Replace x with 0

*Inverse of mult. by -3 is div. by -3

*y-intercept

y-intercept is (0, -2).

Plug in 1 for x to get a third solution:

*Replace x with 1

*Inverse of add 2 is sub. 2

*Inverse of mult. by -3 is div. by -3

(1, -4/3) is another solution on the boundary line.

Solutions:

x	y	*(x, y)*
3	0	(3, 0)
0	-2	(0, -2)
1	-4/3	(1, -4/3)

Since the original problem has a <, this means it DOES include the boundary line.

So are we going to draw a solid or a dashed line for this problem?
It looks like it will have to be a solid line.

Putting it all together, we get the following boundary line for this problem:

example 2e

Step 2: Plug in a test point that is not on the boundary line.

Note how the boundary line separates it into two parts.

An easy test point would be (0, 0). Note that it is a point that is not on the boundary line. In fact, it is located above the boundary line.

Let’s put (0, 0) into the original problem and see what happens:

*Replace x and y with 0
*True statement

Step 3: Shade in the answer to the inequality.

Since our test point (0, 0) made our inequality TRUE, this means it is a solution.

Our solution would lie above the boundary line. This means we will shade in the part that is above it.

Note that the gray lines indicate where you would shade your final answer.

Example 3: Graph the inequality

Step 1: Graph the boundary line.

If we wrote this as an equation, it would be x = 4. This is in the form x = c, which is one of our “special” lines.

Do you remember what type of line x = c graphs as?
It comes out to be a vertical line.
If you need a review on vertical lines, go to Tutorial 14: Graphing Linear Equations.

Every x’s value on the boundary line would have to be 4.

Solutions:

x	y	(x, y)
4	0	(4, 0)
4	1	(4, 1)
4	2	(4, 2)

Since the original problem has a <, this means it DOES NOT include the boundary line.

So are we going to draw a solid or a dashed line for this problem?
It looks like it will have to be a dashed line.

Putting it all together, we get the following boundary line for this problem:

example 3b

Step 2: Plug in a test point that is not on the boundary line.

Note how the boundary line separates it into two parts.

An easy test point would be (0, 0). Note that it is a point that is not on the boundary line. In fact, it is located to the left of the boundary line.

Let’s put (0, 0) into the original problem and see what happens:

*Replace x with 0
*True Statement

Step 3: Shade in the answer to the inequality.

Since our test point (0, 0) made our inequality TRUE, this means it is a solution.

Our solution would lie to the left of the boundary line. This means we will shade in the part that is to the left of it

Note that the gray lines indicate where you would shade your final answer.

Graphing the Union
of Two Linear Inequalities

The key word for union is “or”.

This means we will put both graphs on the same coordinate system and shade in the area that includes anything that was shaded in one graph or the other or both. Basically you are putting the two inequalities onto one graph, kind of like having a "+" sign in between the two inequalities..

Note that when you do graph each individual inequality you follow the exact same steps as shown in the example above. The only difference is that you have to do it twice in one problem.

Example 4: Graph the union

Graph the First Inequality

Step 1: Graph the boundary line.

I’m going to use the intercepts to help me graph the boundary line. Again, you can use any method that you want, unless the directions say otherwise.

What value is y on the x-intercept?
If you said 0, you are correct.
If you need a review on x-intercepts, go to Tutorial 14: Graphing Linear Equations.

*Replace x with 0

*Inverse of mult. by 3 is div. by 3

*x-intercept

x-intercept is (5, 0)

What is the value of x on the y-intercept?
If you said 0, you are correct.
If you need a review on y-intercepts, go to Tutorial 14: Graphing Linear Equations.

*Replace x with 0

*Inverse of mult. by 5 is div. by 5

*y-intercept

y-intercept is (0, 3).

Plug in 1 for x to get a third solution:

*Replace x with 1

*Inverse of add 3 is sub. 3

*Inverse of mult. by 5 is div. by 5

(1, 12/5) is another solution on the boundary line.

Solutions:

x	y	*(x, y)*
5	0	(5, 0)
0	3	(0, 3)
1	12/5	(1, 12/5)

Since the original problem has a >, this means it DOES include the boundary line.

So are we going to draw a solid or a dashed line for this problem?
It looks like it will have to be a solid line.

Putting it all together, we get the following boundary line for this problem:

example 4f

Step 2: Plug in a test point that is not on the boundary line.

Note how the boundary line separates it into two parts.

An easy test point would be (0, 0). Note that it is a point that is not on the boundary line. In fact, it is located below the boundary line.

Let’s put (0, 0) into the original problem and see what happens:

*Replace x and y with 0
*False statement

Step 3: Shade in the answer to the inequality.

Since our test point (0, 0) made our inequality FALSE, this means it is not a solution.

Since (0, 0) was below the line and we need to be on the other side, our solution lies above the boundary line. This means we will shade in the part that is above it.

Note that the gray lines indicate where you would shade your final answer.

Graph the Second Inequality

Step 1: Graph the boundary line.

I’m going to use the intercepts to help me graph the boundary line. Again, you can use any method that you want, unless the directions say otherwise.

What value is y on the x-intercept?
If you said 0, you are correct.
If you need a review on x-intercepts, go to Tutorial 14: Graphing Linear Equations.

*Replace y with 0
*x-intercept

x-intercept is (5, 0)

What is the value of x on the y-intercept?
If you said 0, you are correct.
If you need a review on y-intercepts, go to Tutorial 14: Graphing Linear Equations

*Replace x with 0

*Inverse of mult. by -1 is div. by -1

*y-intercept

y-intercept is (0, -5).

Plug in 1 for x to get a third solution:

*Replace x with 1
*Inverse of add 1 is sub. 1

*Inverse of mult. by -1 is div. by -1

(1, -4) is another solution on the boundary line.

Solutions:

x	y	*(x, y)*
5	0	(5, 0)
0	-5	(0, -5)
1	-4	(1, -4)

Since the original problem has a <, this means it DOES include the boundary line.

So are we going to draw a solid or a dashed line for this problem?
It looks like it will have to be a solid line.

Putting it all together, we get the following boundary line for this problem:

example 4l

Step 2: Plug in a test point that is not on the boundary line.

Note how the boundary line separates it into two parts.

An easy test point would be (0, 0). Note that it is a point that is not on the boundary line. In fact, it is located above the boundary line.

Let’s put (0, 0) into the original problem and see what happens:

*Replace x and y with 0
*True statement

Step 3: Shade in the answer to the inequality.

Since our test point (0, 0) made our inequality TRUE, this means it is a solution.

Our solution lies above the boundary line. This means we will shade in the part that is above it.

Note that the gray lines indicate where you would shade your final answer.

The union of the two inequalities is the area on the graph that was shaded in either the first inequality OR the second one OR both.

This is what we get when we union these two inequalities:

Note that the gray lines indicate where you would shade your final answer.

example 4o

Graphing the Intersection
of Two Linear Inequalities

The key word for intersection is “and”.

This means in the end, we only want to shade in the area that satisfies BOTH linear inequalities. We will graph each one separately and then assess what they have in common, this will be our final answer.

Note that when you do graph each individual inequality you follow the exact same steps as shown in the example above. The only difference is that you have to do it twice in one problem.

Example 5: Graph the intersection

AND

Graph the First Inequality

Step 1: Graph the boundary line.

If we wrote this as an equation, it would be x = -3. This is in the form x = c, which is one of our “special” lines.

Do you remember what type of line x = c graphs as?
It comes out to be a vertical line.
If you need a review on vertical lines, go to Tutorial 14: Graphing Linear Equations.

Every x’s value on the boundary line would have to be -3.

Solutions:

x	y	(x, y)
-3	0	(-3, 0)
-3	1	(-3, 1)
-3	2	(-3, 2)

Since the original problem has a >, this means it DOES include the boundary line.

So are we going to draw a solid or a dashed line for this problem?
It looks like it will have to be a solid line.

Putting it all together, we get the following boundary line for this problem:

example 5c

Step 2: Plug in a test point that is not on the boundary line.

Note how the boundary line separates it into two parts.

An easy test point would be (0, 0). Note that it is a point that is not on the boundary line. In fact, it is located to the right of the boundary line.

Let’s put (0, 0) into the original problem and see what happens:

*Replace x with 0
*True Statement

Step 3: Shade in the answer to the inequality.

Since our test point (0, 0) made our inequality TRUE, this means it is a solution.

Our solution would lie to the right of the boundary line. This means we will shade in the part that is to the right of it

Note that the gray lines indicate where you would shade your final answer.

Graph the Second Inequality

Step 1: Graph the boundary line.

If we wrote this as an equation, it would be y = 4. This is in the form y = c, which is one of our “special” lines.

Do you remember what type of line y = c graphs as?
It comes out to be a horizontal line.
If you need a review on horizontal lines, go to Tutorial 14: Graphing Linear Equations.

Every y’s value on the boundary line would have to be 4.

Solutions:

x	y	(x, y)
0	4	(0, 4)
1	4	(1, 4)
2	4	(2, 4)

Since the original problem has a <, this means it DOES NOT include the boundary line.

So are we going to draw a solid or a dashed line for this problem?
It looks like it will have to be a dashed line.

Putting it all together, we get the following boundary line for this problem:

example 5f

Step 2: Plug in a test point that is not on the boundary line.

Note how the boundary line separates it into two parts.

An easy test point would be (0, 0). Note that it is a point that is not on the boundary line. In fact, it is located below the boundary line.

Let’s put (0, 0) into the original problem and see what happens:

*Replace y with 0
*True statement

Step 3: Shade in the answer to the inequality.

Since our test point (0, 0) made our inequality TRUE, this means it is a solution.

Our solution would lie below the boundary line. This means we will shade in the part that is below it

Note that the gray lines indicate where you would shade your final answer.

The intersection of the two inequalities is the area on the graph that was shaded in BOTH the first inequality AND the second inequality. It is the region where they overlap

This is what we get when we intersect these two inequalities:

Note that the gray lines indicate where you would shade your final answer.

example 5i

Practice Problems

These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.