Graphing Linear Inequalities

This is a graph of a linear inequality:

Graph with shaded region below the solid line y = x + 2
The inequality y ≤ x + 2

We 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

Graph the "equals" line, then shade in the correct area.

Follow these steps:

Rearrange the equation so "y" is on the left and everything else on the right
Remember to flip the inequality sign if we multiply or divide by a negative number while rearranging
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):

Solid line graph of y = 2x - 1

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

Graph with shaded region below the solid line y = 2x - 1

Example: 2y − x ≤ 6

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

Start with: 2y − x ≤ 6

Add x to both sides: 2y ≤ x + 6

Divide all by 2: y ≤ x2 + 3

2. Now plot y = x2 + 3 (as a solid line because y≤ includes equal to):

Solid line graph of y = x/2 + 3

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

Graph with shaded region below the solid line y = x/2 + 3

Example: y/2 + 2 > x

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

Start with: y/2 + 2 > x

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):

Dashed line graph of y = 2x - 4

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

Graph with shaded region above the dashed line y = 2x - 4

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

Check With a Point

We can check if we shaded the right side by picking a point (like 0,0) and seeing if it works in the inequality.

In our last example y > 2x − 4:

Try (0,0):

0 > 2(0) − 4

0 > −4

That is True! So (0,0) should be in the shaded area. Looking at the graph, it is!

Two Special Cases

We can also have a horizontal, or vertical, line:

Graph with shaded region below the dashed horizontal line y = 4

This shows where y is less than 4
(from, but not including, the line y=4 on down)

Notice we have a dashed line to show it does not include y=4

Graph with shaded region right of the solid vertical line x = 1

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

Tip: For inequalities like x ≥ 1 we shade right of the vertical line. For x ≤ 1 we shade left.

7426, 7427, 7428, 7429, 7430, 7431, 7432, 7433, 7434, 7435