Graph System of Linear Inequalities Calculator
Graph multi-constraint systems of 2D linear inequalities on an interactive Cartesian plane. Visualize shaded half-planes, solid and dashed boundary lines, feasible solution regions, and corner point vertices for linear programming optimization.
System Inequalities (2 to 5)
Standard syntax: y ≤ 2x + 1, 2x + 3y > 6, x ≥ 0, y < 5. Use ≤, ≥, <, > (or <=, >=).
Interactive Test Point Verification
Verify whether an arbitrary Cartesian point (x, y) satisfies all constraints or falls outside the solution set.
Graph Layers & Styling
Feasible Solution Set Analysis
| Vertex | Coordinates (x, y) | Intersecting Boundaries | Status |
|---|
Foundations of Linear Inequalities & Cartesian Half-Planes
In Cartesian coordinate geometry, a single linear equation in two variables, ax + by = c (where a and b are not both zero), defines a straight one-dimensional line that bisects the continuous two-dimensional plane ℝ² into two infinite, disjoint regions known as half-planes.
When an equality sign is replaced by an inequality relation (≤, ≥, <, >), the mathematical expression no longer represents a one-dimensional curve, but rather an entire two-dimensional half-plane of coordinate points that satisfy the inequality condition:
A system of linear inequalities consists of a set of k simultaneous linear inequality conditions:
The geometric solution to such a system is the set-theoretic intersection of the individual half-planes: Ω = H_1 ∩ H_2 ∩ … ∩ H_k. This intersection constitutes the feasible region.
Solid vs. Dashed Boundary Lines: Inclusive vs. Strict Operators
The boundary line separates coordinates that satisfy the inequality from those that violate it. The visual convention used to graph this boundary communicates critical topological information regarding whether the boundary itself belongs to the solution set:
Solid Boundary Lines (≤, ≥)
Used when the inequality includes equality (non-strict or inclusive). Any point (x_0, y_0) lying exactly on the line satisfies ax_0 + by_0 = c and is an authentic member of the solution set. In mathematical topology, this represents a closed half-space.
Dashed Boundary Lines (<, >)
Used when the inequality is strict. Points lying directly on the line satisfy ax_0 + by_0 = c, but violate the condition ax_0 + by_0 < c. The dashed line serves as a geometric frontier or barrier that is excluded from the solution set, forming an open half-space.
The Test Point Method: Determining Shading Direction via Origin
Once the boundary line ax + by = c is graphed, one must determine which of the two half-planes contains the solutions. The standard mathematical technique is the Test Point Method:
Algorithmic Steps for the Test Point Method:
- Select a Candidate Coordinate: Choose an arbitrary coordinate pair (x_t, y_t) that does not lie on the boundary line. The origin (0, 0) is the optimal choice because arithmetic evaluation is trivial.
- Substitute Coordinates into Inequality: Evaluate the left-hand algebraic expression a(0) + b(0) = 0. Compare the resulting value against constant c.
- Evaluate Truth Value:
- If the inequality statement is TRUE (e.g., 0 ≤ 6), shade the half-plane that contains the test point (0, 0).
- If the inequality statement is FALSE (e.g., 0 ≥ 8), shade the opposite half-plane on the other side of the boundary line.
- Exception for Lines Passing Through (0, 0): If the boundary line passes through the origin (i.e., c = 0, such as y ≤ 2x), select an alternative test point along a primary axis, such as (0, 1) or (1, 0).
The Feasible Solution Region: Intersection of Half-Planes
When multiple inequalities are plotted on the same coordinate plane, each inequality shades a different portion of Cartesian space. The true solution to the system is not the union of all shaded regions, but strictly their mutual intersection.
Every coordinate point (x, y) lying inside the feasible region simultaneously satisfies inequality 1, inequality 2, and all subsequent constraints. If a point violates even a single constraint, it is immediately discarded from the feasible set.
Bounded Convex Polygon
The feasible region is completely surrounded by intersecting boundary segments, forming a closed polygon with a finite area and a finite set of extreme vertices.
Unbounded Feasible Region
The overlapping region opens up and extends infinitely in one or more directions. Coordinates can grow arbitrarily large while still satisfying all system constraints.
Empty Feasible Set (∅)
The system is inconsistent. The shaded half-planes point in contradictory directions and share zero common points, meaning no solution exists in ℝ².
Linear Programming Connection: Corner Point Theorem & Extrema
Graphing systems of linear inequalities is the foundational graphical method for Linear Programming (LP). In linear programming, decision variables x and y are subject to structural constraints (such as raw materials, labor hours, and non-negativity x ≥ 0, y ≥ 0), while an objective function Z = dx + ey is to be maximized (e.g., profit) or minimized (e.g., cost).
The Corner Point (Extreme Point) Theorem
If the feasible region of a system of linear inequalities is non-empty and bounded, the linear objective function Z = dx + ey achieves its maximum and minimum values at one or more of the corner points (vertices) of the feasible polygon.
Because level curves of the objective function are parallel straight lines dx + ey = c, moving the level line across the convex feasible polygon will always exit the region at one of its extreme vertices or along an entire boundary edge connecting two optimal vertices.
Step-by-Step Algorithm for Graphing Systems of Inequalities
Express Constraints in Boundary Form
Convert each inequality to its corresponding linear equation a_i x + b_i y = c_i. Isolate y into slope-intercept form y = mx + b when convenient, or determine intercept coordinates (c/a, 0) and (0, c/b).
Plot Boundary Lines with Correct Line Stroke
Draw a solid continuous line if the operator is ≤ or ≥. Draw a dashed segmented line if the operator is strict (< or >).
Determine Half-Plane Shading for Each Constraint
Substitute the origin (0, 0) into each inequality. Shade the side containing (0, 0) if the expression evaluates to true; shade the opposite side if false.
Locate Feasible Region Intersection
Highlight the overlapping region common to all shaded half-planes. If any inequality does not overlap with the others, the intersection is empty.
Solve for Feasible Vertices (Corner Points)
Set up and solve simultaneous 2 × 2 linear systems for each pair of intersecting boundary lines. Verify that each calculated intersection coordinate satisfies all remaining inequalities in the system.
Systems Classification Table: Bounded, Unbounded & Inconsistent
The table below contrasts the three fundamental topological states of systems of linear inequalities:
| System Type | Geometric Topology | Vertex Count | LP Optimization Behavior | Archetype Example |
|---|---|---|---|---|
| Bounded Feasible Polygon | Enclosed convex polygon; finite surface area | ≥ 3 vertices | Both max & min guaranteed to exist | x ≥ 0, y ≥ 0, x+2y ≤ 8, 3x+2y ≤ 12 |
| Unbounded Cone / Half-Space | Infinite wedge extending without upper limit | ≥ 1 vertex | Minima may exist; maxima may diverge to ∞ | x ≥ 0, y ≥ 0, x + y ≥ 4 |
| Inconsistent / Empty Set (∅) | No common intersection points exist | 0 vertices | Infeasible; no solution possible | y ≥ x + 4, y ≤ x - 2 |
Graded Worked Problems with Complete Solutions
Graph the system of inequalities and find all corner point vertices of the feasible region:
1) x ≥ 0 2) y ≥ 0 3) x + 2y ≤ 8 4) 3x + 2y ≤ 12
Complete Step-by-Step Solution:
Step 1: Non-Negativity Constraints.
x ≥ 0 restricts the feasible set to the right of the vertical y-axis. y ≥ 0 restricts the set above the horizontal x-axis. Together, these bound the solution to the First Quadrant.
Step 2: Boundary Line 3 (x + 2y = 8).
Find intercepts: For x = 0 ⇒ y = 4, coordinate is (0, 4). For y = 0 ⇒ x = 8, coordinate is (8, 0).
Test (0, 0): 0 + 2(0) ≤ 8 ⇒ 0 ≤ 8 (True). Shade below and toward the origin.
Step 3: Boundary Line 4 (3x + 2y = 12).
Find intercepts: For x = 0 ⇒ y = 6, coordinate is (0, 6). For y = 0 ⇒ x = 4, coordinate is (4, 0).
Test (0, 0): 3(0) + 2(0) ≤ 12 ⇒ 0 ≤ 12 (True). Shade below and toward the origin.
Step 4: Solve for the Interior Intersection Vertex.
Subtract line 3 from line 4:
(3x + 2y) - (x + 2y) = 12 - 8 ⇒ 2x = 4 ⇒ x = 2.
Substitute x = 2 into line 3: 2 + 2y = 8 ⇒ 2y = 6 ⇒ y = 3.
Intersection vertex is (2, 3).
Step 5: Compile Feasible Vertices.
The feasible region is a bounded quadrilateral with 4 vertices:
V1 = (0, 0) • V2 = (4, 0) • V3 = (2, 3) • V4 = (0, 4).
Determine the solution set for the system:
1) y < 2x + 4 2) y > 2x - 2
Diagnostic Explanation:
Both boundary lines have slope m = 2 and are strictly parallel. Because both operators (< and >) are strict, both boundary lines must be drawn as dashed lines.
For y < 2x + 4: Testing (0, 0) ⇒ 0 < 4 (True). Shade below the upper boundary.
For y > 2x - 2: Testing (0, 0) ⇒ 0 > -2 (True). Shade above the lower boundary.
The feasible region is the infinite open parallel diagonal channel trapped between the two dashed lines. It has 0 vertices because parallel lines never intersect.
Real-World Applications in Economics, Operations & Logistics
Manufacturing & Production Mix
A factory producing two products x and y must balance machine fabrication hours, assembly labor, and warehousing capacity. Inequalities outline the allowable daily production combinations.
Dietary & Nutritional Planning
Dietitians design meal regimens satisfying minimum daily allowances for protein, vitamins, and minerals while restricting maximum sodium and saturated fats. The feasible region contains all nutritionally viable diets.
Supply Chain & Cargo Freight
Cargo aircraft operate under strict maximum gross takeoff weight and total volume capacity limits. Linear systems define maximum payloads for high-density versus low-density shipping freight.
Common Pitfalls & Mistakes to Avoid
1. Forgetting to Reverse Inequality Signs When Dividing by Negatives
When converting standard form -2y ≤ 6 - 4x to slope-intercept form, dividing by -2 mandates flipping the inequality symbol: y ≥ 2x - 3. Forgetting to flip the sign causes the wrong half-plane to be shaded, completely invalidating the feasible region.
2. Using (0, 0) as a Test Point for Lines Passing Through Origin
If an inequality has no constant term (e.g., y ≤ 3x), the boundary line passes directly through (0, 0). Substituting (0, 0) yields 0 ≤ 0, which gives no information on which side to shade. Always choose an off-line test point such as (0, 1) or (1, 0).
3. Including Extraneous Boundary Intersections as Vertices
When k boundary lines intersect pairwise, they produce up to k(k-1)/2 intersection points. However, only intersections that satisfy all inequalities in the system are legitimate feasible vertices. Our calculator automatically tests every pairwise intersection against all constraints to filter out false vertices.
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