Graphical Representation of Inequalities

A graph like y equals x squared forms a curved boundary called a parabola, forming two regions on the xy plane.

Replacing the equal sign with an inequality—such as greater than, less than, or equal variations—keeps the curve as a boundary and shades one region to show where the inequality is true.

The inequality includes not just the parabola but also points whose y-coordinates are greater or less than the corresponding values on the curve.

If the inequality uses greater than or equal to, or less than or equal to, the boundary is drawn as a solid curve, indicating those points are included.

If it uses only greater than or less than, the boundary is drawn as a dashed curve, showing those points are excluded.

To identify the correct region to shade, a point from each side of the curve is tested by substituting the coordinates of the point into the inequality. If the point satisfies the inequality, that region is shaded. If not, the opposite side is shaded.

This method is useful in real situations, like plotting a temperature-versus-time graph for a chemical reaction and identifying the time intervals where the temperature stays below a threshold limit by shading the region where this condition is true.