# Graphing Linear Inequalities

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:

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:

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 |