System of Inequalities Calculator

This system of inequalities calculator is designed for the kind of work students usually spread across several tools: graphing multiple inequalities at once, finding the feasible region, checking whether the overlap is bounded or unbounded, identifying corner points, and then applying those same geometry results to a linear programming objective function. The page keeps those tasks in one workflow so the graph and the algebra stay synchronized.

The hero workspace uses one input row per inequality because systems are easier to read when each condition has its own color and line. That makes it clear which boundary belongs to which shaded region. In graph mode, the goal is the feasible region itself. In linear programming mode, the goal becomes an objective value such as max P = 3x + 2y, but the feasible region still comes first because optimization only makes sense after the constraints are understood.

The result area is organized around the decisions students actually need to make. The graph shows the overlap. The Steps tab explains how each inequality is graphed. The Feasible Region tab summarizes the surviving region and its type. The Vertex Solver tab lists every pairwise boundary intersection, validates it against all constraints, and separates valid vertices from discarded candidates.

A system of inequalities is a collection of two or more inequalities that must all be true at the same time. In one variable, the solution is usually an interval or a union of intervals. In two variables, the solution is a region of the coordinate plane rather than a single point.

That is the main difference from a system of equations. A system of equations usually asks for exact intersection points where every equation is satisfied simultaneously. A system of inequalities asks for the set of all points that satisfy every condition, so the answer is typically a shaded overlap called the feasible region.

Systems of inequalities appear in algebra, analytic geometry, economics, operations research, and linear programming. They model constraints: spending cannot exceed a budget, production cannot exceed a machine limit, time cannot exceed a schedule, and variables may need to stay nonnegative. The geometry is not decoration. It is the meaning of the constraint set.

To solve a system of inequalities, graph each inequality separately before you look for the final overlap. Replace each inequality sign with an equal sign to get the matching boundary. Decide whether the boundary is dashed or solid. Then use a test point to choose the side to shade for that one inequality.

After every inequality has its own shaded region, the system solution is the intersection of all those regions. That shared overlap is the feasible region. If no common overlap exists, the system has no solution. If the overlap closes into a polygon, the region is bounded. If it keeps extending forever in some direction, the region is unbounded.

The final algebra step is usually about corner points. For linear systems, corner points come from intersections of boundary lines. Solve boundary equations in pairs, then keep only the intersections that satisfy the full system. Those surviving points matter because they describe the geometry of the feasible region and become the key evaluation points for linear programming.

Graphing a system of inequalities is a repeatable process, not guesswork. For each constraint, draw the boundary line or curve first. Use a dashed boundary for strict inequalities and a solid boundary for inclusive inequalities. Then choose a test point that is not on that boundary and decide which side to shade.

Once every individual inequality has been graphed, the system solution is the part of the plane where all shaded regions overlap. That overlap can be large or small, polygonal or curved, bounded or unbounded. The important point is that a point belongs to the system only if it satisfies every inequality, not just one or two of them.

This page keeps the inequalities color-coded so the graph stays readable when several boundaries overlap. That makes it easier to see why a point satisfies two constraints but fails a third, and it also makes the feasible region easier to distinguish from the individual half-planes around it.

The feasible region is the set of all points that satisfy every inequality in the system. On the graph, it is the overlap of all individual shaded regions. In optimization language, it is the set of all allowable choices under the given constraints.

A feasible region can be bounded, meaning it closes into a polygon or another closed shape with finite area. It can be unbounded, meaning the overlap stretches infinitely in at least one direction. It can also be empty, meaning no point satisfies the full system and the graph has no shared overlap at all.

A quick way to test whether a point is in the feasible region is to substitute its coordinates into every inequality. If every statement is true, the point belongs to the feasible region. If even one statement is false, the point is outside the system solution.

A bounded feasible region is enclosed. In linear systems, that usually means the constraints form a polygon with finitely many corner points and finite area. Bounded regions are common when nonnegativity constraints are combined with upper bounds such as x + y <= 6.

An unbounded feasible region is still a valid solution set, but it keeps extending infinitely in at least one direction. Many two-line systems are unbounded because the overlap creates a wedge or corridor rather than a closed polygon. Unbounded does not mean no solution. It means the solution set never closes off completely.

This distinction matters most in linear programming. If the feasible region is bounded, a linear objective has both a maximum and a minimum on the region. If the feasible region is unbounded, one of those extremes may fail to exist because the objective can keep increasing or decreasing along an open direction.

Corner points, also called vertices, are the places where boundary lines meet to form the edges of a feasible region. In a linear system, they come from solving pairs of boundary equations simultaneously.

The standard workflow is to list every boundary line, solve every pair, and then test each candidate point in the full system. Only the points that survive every constraint count as useful corner points. For n linear inequalities, there are at most n(n-1)/2 boundary pairs to test.

Corner points matter because they compress the geometry of the region into a finite set of strategically important locations. Even when the feasible region is large, the vertices summarize where the edges meet and where a linear objective is most likely to reach its extreme values.

A system of inequalities has no solution when the shaded regions do not overlap at all. This often happens when two parallel lines are shaded away from each other or when the constraints directly contradict one another, such as x > 5 and x < 3.

On the graph, no-solution systems are easy to recognize once each inequality has been shaded correctly. You may still see each individual region, but there is no common feasible region where all the conditions agree.

Algebraically, a no-solution system means there is no ordered pair that makes every inequality true at once. The empty feasible region is written as ∅, and in linear programming it means the optimization problem is infeasible.

Linear programming is the practice of optimizing a linear objective function subject to inequality constraints. A typical problem asks you to maximize or minimize an expression such as P = 3x + 2y while also satisfying a system such as x + y <= 6, x >= 0, and y >= 0.

The crucial theorem is that if an optimal value exists for a linear programming problem, it occurs at a corner point of the feasible region. That is why graphing the constraints and finding the vertices comes before evaluating the objective. Once the feasible region is known, the optimization problem becomes finite: evaluate the objective at the relevant corner points and compare the values.

If the feasible region is bounded, both a maximum and a minimum exist. If the feasible region is unbounded, an objective may still have a minimum or maximum at a corner point, but the other direction can fail to exist because the value keeps improving along an unbounded ray. This page checks the corner points and the open directions together so the optimization result is easier to interpret.