Understanding Graphical Inequalities

Inequalities represent the relationships between variables with a range of possible values. They’re often used in mathematics and everyday life to exhibit a range of possible solutions.
An inequality uses symbols (<, >, ≤, ≥) to show that something is not equal to something else. For example, x > 2 means x is greater than 2.
Graphical inequalities involve sketching the region of a graph that satisfies a given inequality. This will often result in a shaded region on a Cartesian plane.

Drawing Inequalities on Graphs

To graph an inequality, start by drawing a boundary line. The equation for the boundary line is the same as the inequality, but with an equal sign instead. In a gcse question, you are usually given a line to draw or it requires straight lines and can be easily plotted using y=mx+c knowledge from straight line graphs.
The boundary line will be solid lines if values on the line are included in the solution (e.g. y ≤ 2x + 3). They will be dotted lines if values on the line are not included in the solution (e.g. y < 2x + 3).
Once the boundary line is drawn, a test point that is not on the line is chosen. The coordinates of the test point are substituted into the inequality. If the inequality is true for the test point, the half of the plane that includes the test point is shaded. If the inequality is false for the test point, the half of the plane that does not include the test point is shaded.
For instance, if plotting y ≤ 2x + 1, draw a solid line where y = 2x + 1. If we use a test point of (0,0), we see that 0 is not less than or equal to 1, so we would shade the area that does not include the point (0, 0), which is above the line.
When shading, always remember that the shaded area is the solution of the inequality.

Sometimes, more than one inequality is considered at a time, these are called compound inequalities.
In this case, you would graph all of them separately at first.
The final solution is the overlapping shaded region of all the individual inequalities.
If there’s no intersection between the inequalities, then there’s no solution to the compound inequality.

Always use a pencil when drawing the graph so you can correct mistakes.
Make sure to choose a suitable scale for your graph to ensure all important points are clear.
Remember to plot each line accurately. If you’re unsure, use a ruler.
The shading of areas represents the possible solutions to the inequality, ensure this is clear in your working.
Manipulate inequalities algebraically when needed for easier graphing.
Frequent practice not only of graphical inequalities, but also graph sketching and algebraic manipulation will help you improve your skill and speed.