What Is a Graphical Inequality?
You already know how to plot a line like on a coordinate grid. An inequality takes this idea further — instead of asking "which points sit exactly on the line?", it asks "which points sit on one side of the line?"
A mathematical statement that compares two expressions using , , , or , whose solution is an entire region of the coordinate plane rather than a single line or point.
When you solve an inequality in one variable (e.g. ), the answer is a set of -values on a number line. When you solve an inequality in two variables (e.g. ), the answer is a set of points — an entire shaded region of the graph.
In MYP Sciences, we use inequalities as models — they help us represent real constraints like safe temperature ranges, maximum doses, or resource limits. This connects directly to our unit's Key Concept of Models: a graphical inequality is a visual model of a boundary condition in the real world.
Think of the line as a border between two countries. The equation tells you where the border is. The inequality tells you which country satisfies your condition — but both countries still exist on the map.
ATL Connection — Thinking Skills: Graphical inequalities train you to move between symbolic (algebraic) and visual (graphical) representations of the same idea — a key skill in scientific modelling.
Solving Linear Inequalities in One Variable Graphically
Before we tackle two-variable inequalities, it helps to see the idea in one dimension. Consider solving graphically.
Method:
- Draw the two functions: and on the same axes. Draw both as solid lines — they are just functions you are comparing.
- Find their intersection point — this is the value of where both sides are equal.
- The intersection is at (since ).
- Mark with an open circle on the -axis (because the inequality is strict: , so is not included).
- Read off the region where is below , meaning .
Example: Solve graphically.
Draw (gradient , -intercept ) and (a horizontal line).
They intersect where , giving .
For , the line is below , meaning .
Solution:
On a number line, mark an open circle at and shade to the left.
This method builds the intuition you need for two-variable inequalities: you are always asking "on which side of the boundary is one expression smaller/larger than the other?"
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