How to Graph Inequalities With 2 Variables

By Nicole Harms
The right tools will help you solve systems of linear equations accurately.
sanja gjenero

You open your math book and see this problem: “please graph y > 3x +2.” Your hands start to feel clammy as you realize that the problem contains not one, but two variables. How are you supposed to graph an inequality that contains two variables? What do you do with that greater than sign anyway? Before you panic, take the problem apart step by step. Graphing inequalities with two variables is not impossible if you know the right steps to take.

How to Graph Inequalities With 2 Variables

Look at the original inequality. Make sure it is in slope-intercept form (y = mx + b). Keep in mind that you will not have an equals sign, but rather you will have a < or > sign.

Begin by subtracting 2x from both sides to get - 3y < -2x +6. Notice that you put the -2x before the + 6 to get the problem as close to slope-intercept form as possible. If the problem is not in the correct form, you need to change it. For example, 2x – 3y < 6 is not in the correct form.

Divide both sides by -3. In an inequality, when you divide or multiply by a negative number, you must reverse the inequality sign. This gives you y > 2/3x - 2

Plot two points on your graph paper. Start by plotting the y-intercept, which is the number that holds the "b" position in the formula (y = mx + b). In this example the y intercept is -2, so plot (0 , -2).

Identify the slope. Slope, which is the number holding the "m" position in the formula, is expressed as a ratio of rise/run. This means that in this example, the slope is 2/3. From your y intercept you will rise (move up) 2 and run (move to the right) 3.

Apply the slope (up 2, over 3) and plot this point. In this example you will be placing a point at (3 , 0).

Draw a line through the two points you have plotted. If the inequality does not have an "equals to" sign, then you will draw a dashed line indicating that only those values up to the line make the statement true. If the inequality does have the "equals to" sign, draw a solid line.

Finish graphing the inequality by shading. Put the origin (0 , 0) into the original equation. For example, 2 (0) – 3 (0) < 6. Solve this problem (0 < 6). If the statement is true, as in this example, shade the graph on the side of the line that contains the origin. If the statement is false, shade the half of the graph on the side of the line that does not contain the origin.

About the Author

Nicole Harms has been writing professionally since 2006, specializing in real estate, finance and travel. When she's not writing, she enjoys traveling and has visited several countries, including Israel, Spain, France and Guam. Harms received a Bachelor of Science in Education from Maranatha Baptist Bible College.