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System of Inequalities Grapher

Feasible Region · Corner Points · Linear Programming · Up to 5 Constraints

Enter up to 5 linear inequalities, graph all boundary lines, shade each feasible half-plane, and instantly see the combined feasible region with labeled corner points. Perfect for linear programming optimization.

Presets

Inequalities

Use <=, >=, <, > symbols

Graph

Scroll to zoom · Drag to pan · Hover for coordinates

What Is a System of Inequalities?

A system of inequalities is a collection of two or more inequalities involving the same variables that must all be satisfied at the same time. Unlike a system of equations — which has a finite set of discrete solutions — a system of inequalities typically defines a region in the coordinate plane called the feasible region. Every point inside that region satisfies all the constraints simultaneously.

Systems of inequalities arise naturally in real-world problems where you have multiple competing constraints. A manufacturer might be limited by the number of labor hours available (2x + y ≤ 40), the capacity of a machine (x + 3y ≤ 45), and the requirement to produce at least some of each product (x ≥ 0, y ≥ 0). The set of all production plans that respect every constraint is exactly the feasible region of this system.

Graphically, each linear inequality in two variables divides the plane into two half-planes separated by a boundary line. The boundary line is drawn solid for non-strict inequalities (≤, ≥) since boundary points are included, and dashed for strict inequalities (<, >) since boundary points are excluded. The feasible region for a single inequality is one half-plane; for a system, it is the intersection of all the individual half-planes.

How to Use This Grapher

Enter each inequality in the input fields. Use standard notation: y >= 2x + 1, 3x + 2y <= 12, x >= 0. Type <= for ≤ and >= for ≥.
Click "Graph System" to plot all boundary lines, shade individual feasible regions, and highlight the combined feasible region (darker green overlay).
Read the corner points from the table below the graph. Each vertex is the intersection of two boundary lines that also lies inside all constraints.
Test a Point using the checker: enter any (x, y) coordinate and confirm whether it satisfies all inequalities simultaneously.
Zoom and pan by scrolling on the canvas or dragging to explore unbounded regions or see the full picture.

Corner Points and Why They Matter

The vertices (corner points) of the feasible region are the points where two or more boundary lines intersect within the feasible region. They are found algebraically by solving every pair of boundary line equations simultaneously and then checking each candidate point against all constraints.

Corner points are central to linear programming: the Fundamental Theorem of Linear Programming guarantees that if an optimal solution to a linear objective function exists over a bounded feasible region, it is achieved at one of the corner points. This means you can maximize or minimize any linear expression such as P = 5x + 3y by simply evaluating it at each corner point and picking the best value — no calculus required.

Linear Programming — A Quick Primer

Linear programming (LP) is an optimization method used throughout business, engineering, logistics, and economics. An LP problem consists of:

Decision variables (x, y): the quantities you control.
Objective function (e.g., maximize P = 5x + 3y): the quantity to optimize.
Constraints: linear inequalities the variables must satisfy (resource limits, non-negativity, etc.).

The graphical LP method — which this tool supports — works best for two-variable problems. Plot all constraints, identify the feasible region, find every corner point, then evaluate the objective function at each corner to find the maximum or minimum.

Worked Example

Consider the system: x ≥ 0, y ≥ 0, 2x + y ≤ 8, x + 2y ≤ 7.

x >= 0
y >= 0
2x + y <= 8
x + 2y <= 7

Boundary intersections to check:

Boundary Lines	Intersection	Satisfies All?
x = 0 & y = 0	(0, 0)	Yes → corner point
x = 0 & x + 2y = 7	(0, 3.5)	Yes → corner point
y = 0 & 2x + y = 8	(4, 0)	Yes → corner point
2x + y = 8 & x + 2y = 7	(3, 2)	Yes → corner point

The four corner points are (0,0), (4,0), (3,2), (0,3.5). If the objective is to maximize P = 3x + 4y, evaluate at each: P(0,0)=0, P(4,0)=12, P(3,2)=17, P(0,3.5)=14. The maximum is P = 17 at (3, 2). Load this example using the "Basic LP" preset above.

Frequently Asked Questions

What is a feasible region in a system of inequalities?

The feasible region is the set of all points (x, y) that satisfy every inequality in the system simultaneously. On a graph, each inequality shades one half-plane; the feasible region is the overlap — the area where all shaded regions intersect. If no overlap exists, the system has no solution and the feasible region is empty.

How do you find the corner points of a feasible region?

Corner points are found by solving each pair of boundary line equations simultaneously. Take every combination of two boundary lines, solve the 2×2 linear system to find their intersection, then check whether that intersection satisfies ALL inequalities. Points that pass the check are corner points of the feasible region.

What happens if there is no feasible region?

When the inequalities contradict each other — for example x ≥ 5 and x ≤ 2 — no point can satisfy all constraints at once. The shaded regions do not overlap, so there are no corner points and the feasible region is empty (infeasible). In linear programming this means the problem has no solution under the given constraints.

What is linear programming and why are corner points important?

Linear programming is an optimization technique that maximizes or minimizes a linear objective function (e.g., profit = 5x + 3y) subject to linear inequality constraints. The Fundamental Theorem of Linear Programming states that the optimal value always occurs at a corner point of the feasible region. So you only need to evaluate the objective at each corner and compare — no calculus required.

How does the grapher decide which side of a line to shade?

For each inequality the parser rewrites it in the form y OP mx + b (or handles vertical lines x OP c). It then shades the half-plane where the inequality holds: for y < mx + b it shades below, for y > mx + b above. Strict inequalities (<, >) use a dashed boundary line; non-strict (≤, ≥) use a solid line. The combined feasible region is drawn with a darker overlay where all individual regions overlap.

What is the difference between this tool and a single inequality solver?

A single inequality solver (like the Inequality Solver page) finds the algebraic solution set of one inequality — an interval on a number line. This tool handles systems of two or more inequalities in two variables (x and y), graphing each as a half-plane and finding their intersection in 2D space. It is designed for linear programming and 2D feasibility analysis, not 1D algebraic solutions.

What formats can I enter inequalities in?

You can enter inequalities in slope-intercept form (y > 2x + 1), standard form (3x + 2y <= 12), or as simple bounds (x >= 0, y >= 0). Type <= for ≤ and >= for ≥. Both integers and decimals work in coefficients. The parser handles all four inequality symbols.

Can the feasible region be unbounded?

Yes. If the constraints do not form a closed polygon — for example, only x ≥ 0 and y ≥ 0 with no upper bounds — the feasible region extends infinitely in some direction. The graph shows the visible portion within the current view window. For linear programming, an unbounded feasible region may still have a finite optimal solution if the objective function is bounded in the optimization direction.