Graphing Linear Inequalities

In this lesson, we will learn how to solve inequalities using graphs.

In the following diagram:
Aall 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 y ≥ x – 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.

The following videos show more examples of graphing linear inequalities.

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