How to Graph Linear Inequalities

By Stephanie Ellen; Updated April 24, 2017
The process of graphing inequalities is very similiar to that of graphing linear equations.

A linear equation is an equation that makes a line when graphed. A linear inequality is the same type of expression with an inequality sign rather than an equals sign. For example, the general formula for a linear equation is y = mx + b, where m is the slope and y is the intercept. The inequality y < mx + b means that instead of y being equal to mx + b, y is less than mx + b. In an inequality, y is a range of numbers instead of one specific number.

Replace the inequality sign with an equals sign. For example, y>2x becomes y=2x.

Make a table of values by solving your equation for at least two values of x. You could solve your equation for more than two values of x, but you need a minimum of two points to draw a straight line. For example, if you are graphing the equation y=2x, you can substitute x with, say, the numbers 1 and 10:

y=2(1)=2 y=2(10)=20

Draw X and Y axes on your graph paper with the pencil and ruler. The X axis runs across the center of the paper, and the Y axis runs up and down the middle. The graph looks like a cross.

Draw the first point from Step 2 on the graph, where you solved for x=1 and got y=2. That gives you the ordered pair (1,2). Count one space to the right of the center of the graph and two spaces up. Put a dot at that point with your pencil.

Graph the second point from Step 2. Use the same method described in Step 4 to place a dot at (2,20).

Connect the two dots with a ruler and pencil to form a a straight line. This is the graph of your equation.

Shade the graph according to your original inequality in Step 1. For example, y>2x means "y is greater than 2x." In other words, the solutions to the inequality include all numbers larger than those on your graphed line. Here, larger means more positive on the number line, so shade the area to the right of the graphed line with the pencil. If your original inequality had instead used the "less than" symbol, you would be shading to the left of the line.

Tip

Choose one x value that is low and one that is high when making your table of values. The standard coordinate graph has x values from -10 to 10. Therefore, when choosing your x values to graph the equation, pick 1 and 10. These numbers are easy to work with, and they are spread apart so that your graph will be more accurate. If you have a non-standard graph where x runs, for example, from 0 to 150, you'll want to choose x=1 and x=150.

About the Author

Stephanie Ellen teaches mathematics and statistics at the university and college level. She coauthored a statistics textbook published by Houghton-Mifflin. She has been writing professionally since 2008. Ellen holds a Bachelor of Science in health science from State University New York, a master's degree in math education from Jacksonville University and a Master of Arts in creative writing from National University.