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Introduction to graphing inequalities | Two-variable linear inequalities | Algebra I | Khan Academy

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Practicing Algebra 1 math: 'Graph inequalities'

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Fun math practice! Improve your skills with free problems in 'Graph inequalities' and thousands of other practice lessons.

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Graphing inequalities is very similar to graphing linear equations.

Once your linear equation is graphed, you then must focus on the inequality symbol and perform two more steps. It's pretty easy and fun. Stick with me and you'll have no problems by the end of this lesson.

This is a graph for a linear inequality. Notice how we have a boundary line (that can be solid or dotted) and we have a half plane shaded. (Hint: These are the two extra steps that you must take when graphing inequalities.)

In order to succeed with this lesson, you will need to remember how to graph equations using slope intercept form.

There will be two additional steps that you must take when graphing linear inequalities.

As shown in this image, the first step will be to determine whether you will use a solid boundary line or a dashed boundary line. The inequality symbol will help you to determine the boundary line.

If the inequality symbol is greater than or less than, then you will use a dotted boundary line. This means that the solutions are NOT included on the boundary line.

If the inequality symbol is greater than or equal to or less than or equal to, then you will use a solid line to indicate that the solutions are included on the boundary line.

The second step will be to determine which half plane to shade.

If this all sounds confusing, don't worry.... Take a look at the examples below and it will all make sense.

So... first let's take a look at our graphing symbols. You may want to keep this handy for a reference.

Example 1 - Graphing Inequalities

Graph the inequality: y < 2x + 2

Step 1: Graph the inequality as you would a linear equation.

Think of: y = 2x + 2 when you create the graph.

Remember to determine whether the line is solid or dotted. In this case, since the inequality symbol is less than (<), the line is dotted. The points on the line are NOT solutions!

Step 2: Determine which side of the line contains the solutions. Since y is less than the expression, you will shade belo w the line.

If you are unsure of which side to shade, pick any point on the graph (that's not on the line). You are choosing a test point to determine which side contains the solutions.

I will choose (0,0) because this is the easiest point to substitute into the inequality to check for solutions.

Step 3: Substitute (0,0) into the inequality

y < 2x + 2

0 < 2(0) + 2

0 < 2

This is a TRUE statement. 0 is less than 2.

Since (0,0) is a solution and is to the right of the line, ALL of the points to the right of the line are solutions!

Therefore, we will lightly shade the area to the right of the line to show that this side of the line contains all of the solutions to the inequality.

Step 4: Shade the side of the line that contains the solutions to the inequality.

Did you notice how our boundary line was a dotted line because of the less than symbol that was used in the inequality?

Also, you may have realized that you shade below the dotted line because of the less than symbol in the inequality. However, if you are unsure you can always choose a test point. I always use the point (0,0) if it's not on the line.

Substitute (0,0) into the original inequality. If the math sentence is true once you substitute (0,0), then that means that (0,0) is a solution and you shade the half plane that contains (0,0). If the math sentence is false when you substitute (0,0), then that means that (0,0) is not a solution and the other half plane (or the side of the line that does not contain (0,0) should be shaded.

For this second example, we'll need to rewrite the equation so that it's in slope intercept form before we graph. Also take note that the sign is greater than or equal to, so we will graph a solid line this time instead of a dotted line. This example will also demonstrate how to choose three solutions to the inequality.

Example 2: Rewriting the Inequality in Slope Intercept Form.

Graph the following inequality. Then identify three solution to the inequality.

Step 1: We need to rewrite the inequality so that it is in slope intercept form.

Step 2: Graph the line. Note that the line is solid because the inequality sign is greater than or equal to.

Step 3: Shade the solution set. Since y is greater than the expression, shade the side "above" the line.

If you are unsure, pick a "test" point to determine which side of the line you should shade to show the solutions to the inequality.

I am going to choose (0,0) again.

Since 0 is greater than -6, we will shade above the line that since it contains (0,0).

Step 5: Identify three points that are solutions to the inequality. This means that you can pick any three points that are in the shaded area.

Three points that are solutions are:

You may have chosen three completely different solutions, but as long as they are contained in the shaded area, they are solutions to the inequality.

How can you verify that they are solutions?

Substitute the ordered pairs into the inequality!

Let's substitute (1,2)

6 is greater than -4. Since this is a true statement, (1,2) is a solution to the inequality.

I hope this helps you to understand how to graph linear inequalities. If you want to practice a few problems on your own, check out our graphing inequalities practice problems.

Still struggling with inequalities? Check out our other inequality lessons as you continue your studies.

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Graphing inequalities is very similar to graphing linear equations. Once your linear equation is graphed, you then must focus on the inequality symbol and perform two more steps. It's pretty easy and fun. Stick with me and you'll have no problems by the end of this lesson.

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Graphing Linear Inequalities Examples of linear inequalities in two variables: To graph a linear inequality in two variables, first graph the line as if it were a linear equation. Identify whether the inequality includes the "or equal to" aspect.

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This is a graph of a linear inequality:

The inequality y ≤ x + 2

You can see the y = x + 2 line, and the shaded area is where y is less than or equal to x + 2

Linear Inequality

A Linear Inequality is like a Linear Equation (such as y = 2x+1) ...

... but it will have an Inequality like <, >, ≤, or ≥ instead of an =.

How to Graph a Linear Inequality

First, graph the "equals" line, then shade in the correct area.

There are three steps:

Rearrange the equation so "y" is on the left and everything else on the right.

Plot the "y=" line (make it a solid line for y≤ or y≥, and a dashed line for y< or y>)

Shade above the line for a "greater than" (y> or y≥)
or below the line for a "less than" (y< or y≤).

Let us try some examples:

Example: y≤2x-1

1. The inequality already has "y" on the left and everything else on the right, so no need to rearrange

2. Plot y=2x-1 (as a solid line because y≤ includes equal to)

3. Shade the area below (because y is less than or equal to)

Example: 2y − x ≤ 6

1. We will need to rearrange this one so "y" is on its own on the left:

Add x to both sides: 2y ≤ x + 6

Divide all by 2: y ≤ x/2 + 3

2. Now plot y = x/2 + 3 (as a solid line because y≤ includes equal to)

3. Shade the area below (because y is less than or equal to)

Example: y/2 + 2 > x

1. We will need to rearrange this one so "y" is on its own on the left:

Subtract 2 from both sides: y/2 > x − 2

Multiply all by 2: y > 2x − 4

2. Now plot y = 2x − 4 (as a dashed line because y> does not include equals to)

3. Shade the area above (because y is greater than)

The dashed line shows that the inequality does not include the line y=2x-4.

Two Special Cases

You could also have a horizontal or vertical line:

This shows where y is less than 4
(from, but not including, the line y=4 on down)
Notice that we have a dashed line to show that it does not include where y=4

This one doesn't even have y in it!
It has the line x=1, and is shaded for all values of x greater than (or equal to) 1

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This is a graph of a linear inequality: The inequality y ≤ x + 2 You can see the y = x + 2 line, and the shaded area is where y is less than or equal to x + 2 A Linear Inequality is like a Linear Equation (such as y = 2x+1) ...

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When you are graphing inequalities, you will graph the ordinary linear functions just like we done before. The difference is that the solution to the inequality is not the drawn line but the area of the coordinate plane that satisfies the inequality.

The linear inequality divides the coordinate plane into two halves by a boundary line (the line that corresponds to the function). One side of the boundary line contains all solutions to the inequality

The boundary line is dashed for > and < and solid for ≥ and ≤.
Example
Y ≤ 2x - 4

Here you can see that one side is colored grey and the other side is colored white, to determined which side that represent y ≤ 2x - 4, test a point.

We test the point (3;0) which is on the grey side.

$$y\leq 2x-4$$

$$0\leq 2\cdot 3-4$$

$$0\leq 6-4

$$0\leq 2$$

The grey side is the side that symbolizes the inequality y ≤ 2x - 4.

Video lesson

Graph the inequality y > 2 - 2x

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Graphing linear inequalities (Pre-Algebra, Graphing and functions) - Mathplanet

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When you are graphing inequalities, you will graph the ordinary linear functions just like we done before. The difference is that the solution to the inequality is not the drawn line but the area of the coordinate plane that satisfies the inequality.

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IXL | Graph a two-variable linear inequality | Algebra 1 math

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Improve your math knowledge with free questions in “Graph a two-variable linear inequality” and thousands of other math skills.