Graphical Representation of Linear Inequalities in 2 Variables

Graphical Representation of Linear Inequalities in Two Variables

A linear inequality in two variables takes the form of ax + by ≥ c, ax + by ≤ c, ax + by > c, or ax + by < c.
These inequalities can be graphed on a coordinate plane and are represented by a straight line along with a region above or below it.
This forms what we call a half-plane - the area of the graph that represents the solution to the inequality.

To graph a linear inequality, first replace the inequality sign with an equality sign and draw the corresponding line on the graph.
The type of line changes according to the inequality sign. Use a solid line for ≥ or ≤ (indicating that points on the line are included in the solution) and a dashed line for > or < (indicating that points on the line are not included in the solution).
Choose a test point, usually the origin (0,0) unless the line passes through it, and substitute its coordinates into the original inequality. If it satisfies the inequality, shade the half-plane that includes the test point. Otherwise, shade the opposite half-plane.

Graphical solutions of linear inequalities can be used to solve optimisation problems. For example, if we are given certain constraints as linear inequalities, their graphical representation can tell us the feasible region where we could find the optimal solution.
They can also be used to problem-solve in economics and business, where we often want to maximise or minimise something under certain restrictions.

The intersection of the shaded regions (feasible region) from the inequalities represents the solution for a system of linear inequalities.
In some cases, it’s possible to have no solution if the shaded regions don’t intersect.

Don’t forget to flip the inequality sign when multiplying or dividing by negative numbers.
Be careful with the type of line you use when graphing, as including or excluding border points can significantly alter solutions.
Always perform a test check to ensure shading of the correct region.

Illustrate various linear inequalities on a graph: this will help solidify your understanding of the rules and spatial concept.
Practice how moving the inequality changes the boundaries and shaded area: this allows you to predict the probable solutions for a system of linear inequalities.
Run through possible scenarios: solving, graphing, and interpreting linear inequalities can help tackle a variety of problems involving constraints.