How to Graph Inequalities on a Number Line

Number lines are extremely useful for plotting inequalities.
••• BananaStock/BananaStock/Getty Images

The graph of an inequality on a number line can help students visually understand the solution to an inequality. Plotting an inequality on a number line requires a number of rules to ensure the solution is properly “translated” onto the graph. Students should pay special attention to whether the points on the number line are dots or circles, as they represent different types of inequalities.

    Draw the number line. Sketch a long, horizontal line with arrow tips on both ends. Between the arrow tips, add short vertical lines at even intervals along the number line.

    Observe the number in your inequality. For example, if your inequality is “x < 6,” the number of importance is 6. If your inequality has multiple points, such as in “9 < x < 10,” you will have two points of importance.

    Label the vertical lines, or points, on the number line. Label one of the numbers of importance first. Choose a point close to the middle. Label the other points, adding one when going right and subtracting one when going left. Ensure that both points of importance appear on your number line if you have two points of importance.

    Determine the type of point you will need to draw. Look at the sign in the inequality. If your inequality sign does not contain a solid line underneath, you will need to draw an open point, or circle. If you have a line beneath the inequality symbol, you will need to draw a solid point, or dot. If your inequality has two signs, consider each part individually.

    Draw the point or points at the appropriate place or places on the number line.

    Determine whether the inequality is less-than or greater-than. A less-than sign is one that points toward the x, such as in “x < 9.” A greater-than sign is one that points away from the x, such as in “x > 9.” Make this determination for each side of x in an inequality such as “9 < x < 10.”

    Draw an arrow on the number line to indicate an inequality. From the point that you have drawn, draw an arrow to the left if your inequality is a less-than inequality. Draw an arrow to the right if it is a greater-than inequality. Do the same for the other point if you have two points of importance in your inequality. If you have an equation such as "9 < x < 10," you can connect the points with a solid line.