Calculator Inputs
Use up to six two-variable inequalities. The responsive form shows three cards on large screens, two on medium screens, and one on small screens.
Example Data Table
This example matches the default values loaded in the form.
| Item | Example value | Meaning |
|---|---|---|
| Inequality 1 | x + y ≤ 8 | Upper diagonal cap |
| Inequality 2 | x ≥ 1 | Left-side lower bound |
| Inequality 3 | y ≥ 1 | Bottom-side lower bound |
| Inequality 4 | x + 2y ≤ 10 | Second sloped cap |
| Viewing window | 0 ≤ x ≤ 10, 0 ≤ y ≤ 10 | Graph and clipping range |
| Objective | z = 3x + 2y | Vertex comparison rule |
| Test point | (2, 2) | Point feasibility check |
Formula Used
This calculator analyzes a two-variable linear system written in the standard form:
a₁x + b₁y ⊛ c₁, a₂x + b₂y ⊛ c₂, ..., aₙx + bₙy ⊛ cₙ
where each operator ⊛ is one of <=, <, >=, >.
1. Boundary line intersection
Each inequality has a boundary line ax + by = c. Intersections are computed by solving pairs of boundary equations:
a₁x + b₁y = c₁
a₂x + b₂y = c₂
The determinant is:
D = a₁b₂ - a₂b₁
If D ≠ 0, the pair intersects at one point. Every candidate point is then tested against all inequalities.
2. Feasibility test
A point (x, y) is feasible only if it satisfies every active inequality. For example:
ax + by ≤ c
is accepted when the left side does not exceed the right side, within the selected tolerance.
3. Objective function
The objective function is:
z = px + qy
For a bounded feasible polygon, linear objective extremes occur at vertices, so the calculator evaluates z at each feasible vertex.
4. Feasible region area
When the feasible set forms a polygon, area is computed using the shoelace formula:
Area = ½ |Σ(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|
5. Polygon centroid
The centroid gives the balance point of the feasible polygon and is derived from the ordered vertex list.
How to Use This Calculator
- Activate the inequality cards you want to include.
- Enter coefficients for x and y, choose the operator, then enter the constant term.
- Set the viewing window to define graph limits and clipping bounds.
- Enter objective coefficients if you want vertex-based optimization.
- Optionally enter a test point to check whether it satisfies all active inequalities.
- Click Solve System to display the results immediately below the header and above the form.
- Review the plot, vertex table, centroid, area, perimeter, and objective extremes.
- Use the export buttons to download the result summary as CSV or PDF.
Frequently Asked Questions
1. What kind of inequalities does this calculator solve?
It solves linear inequalities in two variables, x and y. Each active rule must have the form ax + by ≤ c, ax + by < c, ax + by ≥ c, or ax + by > c.
2. Why does the graph use a viewing window?
The viewing window keeps the graph readable and lets the calculator clip the region for plotting. Reported area and vertices are based on that visible analysis window.
3. Can this calculator handle strict inequalities?
Yes. It checks strict rules using tolerance-aware comparisons. The graph still draws the boundary line as a guide, even though the exact boundary is excluded mathematically.
4. How are maximum and minimum objective values found?
For a bounded feasible polygon, a linear objective reaches its extreme values at vertices. The calculator evaluates z = px + qy at each feasible vertex and reports the largest and smallest results.
5. What does the signed margin mean for the test point?
Signed margin shows how far the point sits on the allowed side of a constraint. Positive values mean the point satisfies that inequality, while negative values indicate violation.
6. Why might the calculator report no feasible region?
That happens when no point inside the selected viewing window satisfies every active inequality at once. Contradictory constraints are the most common cause.
7. Is the centroid always meaningful?
The centroid is most meaningful when the feasible set forms a polygon with positive area. For a single point or line segment, it becomes a simple average of available feasible points.
8. What do the CSV and PDF exports include?
They include the main summary, objective rule, region type, and feasible vertex data. This makes it easier to archive results, share findings, or document worked examples.