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Graphing Linear Inequalities




 
In these lessons, we will learn how to graph linear inequalities in two variables.

Related Topics:
More Algebra Lessons

In the following diagram:
All the points above the line y = 1 are represented by the inequality y > 1.
All the points below the line are represented by the inequality y < 1. 
The representation is clearer if you look at what the y-coordinates of these points have in common.


In the diagram below, the region above the line is represented by y > 2x –1 and the region below the line is represented by y < 2x – 1.





Example:
By shading the unwanted region, show the region represented by the inequality 2x – 3y ≥ 6

Solution:
First, we need to draw the line 2x – 3y = 6.

We will revise the method for drawing a straight line.

Rewrite the equation in the form y = mx + c.
From the equation m will be the gradient and c will be the y-intercept.
2x – 3y = 6 ⇒  y = x – 2
The gradient is then  and the y-intercept is – 2.

If the inequality is ≤ or ≥ then we draw a solid line. If the inequality is < or > then we draw a dotted line.

After drawing the line, we need to shade the unwanted region.
Rewrite the inequality 2x – 3y ≤ 6 as yx – 2. Since the inequality is ≥, the wanted region is above the line and so the unwanted region is below the line. We shade below the line.



 

Example:
By shading the unwanted region, show the region represented by the inequality x + y < 1

Solution:
Rewrite the equation x + y = 1in the form y = mx + c.
x + y =1 ⇒  y = –x + 1

The gradient is then –1 and the y-intercept is 1.

We need to draw a dotted line because the inequality is <.

After drawing the dotted line, we need to shade the unwanted region.

Rewrite the inequality x + y < 1 as y < –x + 1. Since the inequality is < , the wanted region is below the line and so the unwanted region is above the line. We shade above the line.



Graphing Linear Inequalities Video Lesson
1. Rewrite inequality in slope-intercept form
2. Determine whether to use a solid or a dashed line
3. Determine whether to shade above or below the line.
Graphing Linear Inequalities
In this lesson, you will learn to graph inequalities in two variables.
For example, to graph y > -4x -3, the first step is to graph the boundary line y = -4x -3.
Note that greater than or less than means that the boundary line will be dotted, and greater than or equal to or less than or equal to means that the boundary line will be solid.
To determine which side of the boundary line to shade, substitute a test point, such as (0, 0), into the original inequality, y > -4x -3. Since (0) is greater than (0) -3, or 0 is greater than -3, is a true statement, the side of the line that contains the point (0, 0) is shaded.


 
The following videos show more examples of graphing linear inequalities.



Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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