Calculator Form
Example Data Table
| Inequality | a | b | Operator | c | Meaning |
|---|---|---|---|---|---|
| x + y ≤ 10 | 1 | 1 | ≤ | 10 | Total quantity limit |
| x ≥ 0 | 1 | 0 | ≥ | 0 | Nonnegative x value |
| x - y ≤ 2 | 1 | -1 | ≤ | 2 | Difference limit |
Formula Used
Each inequality is written as:
ax + by < c, ax + by ≤ c, ax + by > c, or ax + by ≥ c
The boundary line uses equality:
ax + by = c
For two boundary lines, intersection points are found with:
D = a1b2 - a2b1 x = (c1b2 - c2b1) / D y = (a1c2 - a2c1) / D
Every candidate point is tested in every inequality. Passing points belong to the visible feasible region.
The optional objective value is:
Z = px + qy
How to Use This Calculator
- Enter each inequality as a coefficient for x, a coefficient for y, an operator, and a constant.
- Use zero for missing x or y terms.
- Set graph limits that include the area you want to inspect.
- Set the grid step for sample point checking.
- Add objective values when you want a visible maximum or minimum.
- Press the calculate button.
- Review the result above the form.
- Download the CSV or PDF report when needed.
Article
Understanding Inequality Systems
A system of linear inequalities describes many half planes on one coordinate plane. Each inequality cuts the plane with a straight boundary line. The answer is the shared region that satisfies every condition at the same time. This calculator focuses on two variable systems, because they are easy to inspect, graph, and explain.
Why Feasible Regions Matter
The feasible region can show possible prices, quantities, schedules, mixtures, or limits. In algebra, it helps students see how symbols become a shaded area. In optimization, it gives the possible choices before a best value is selected. A closed corner can become a maximum or minimum point for a linear objective.
What The Calculator Checks
Each inequality is converted to the form ax plus by compared with c. The calculator tests boundary intersections, box corners, and boundary points inside your chosen graph limits. Points that pass every inequality are listed as feasible vertices. Strict inequalities are treated as open boundaries, so their exact boundary points may be excluded.
Using Limits Carefully
Graph limits do not change the real algebraic answer. They only define the viewing window and the search box. Wide limits are useful for large answers. Smaller limits make the table easier to read. If the region is unbounded, the calculator can still show feasible points inside the selected window.
Reading The Result
A system may have no solution, one narrow ray, a polygon, or an unbounded region. When three or more vertices appear, the ordered table helps describe the visible polygon. When no vertex appears, try wider limits or check whether the inequalities conflict. Always compare the table with the original problem statement.
Practical Uses
Teachers can use the page for examples. Students can verify homework steps. Business users can model simple capacity and budget conditions. The objective option also supports early linear programming practice. Export buttons help keep work for notes, reports, and class records.
Good Habits
Use clear coefficients. Put missing terms as zero. Keep units consistent. Choose a sensible precision level. For strict inequalities, remember that dashed boundary lines are not included. For non strict inequalities, boundary points belong to the solution whenever they pass all other conditions. Then save exports after reviewing every value.
FAQs
What is a system of linear inequalities?
It is a group of linear inequality statements using the same variables. The solution contains all points that satisfy every inequality at once.
What does the feasible region mean?
The feasible region is the shared area that passes all conditions. It can be bounded, unbounded, empty, or partly outside the chosen window.
Can I enter strict inequalities?
Yes. The calculator accepts less than and greater than signs. Their boundary lines are treated as open, so exact boundary points are excluded.
Why are graph limits needed?
Graph limits define the visible search window. They do not change the actual algebraic solution, but they affect displayed points and samples.
What if no vertex appears?
The system may be infeasible, unbounded, or outside the viewing window. Try wider limits and check each inequality for typing mistakes.
Can this solve three variable inequalities?
No. This calculator is designed for two variables. Three variable systems need three dimensional methods and different visualization tools.
What is the objective option?
The objective option tests Z = px + qy. It reports a visible maximum or minimum among available vertices or sample points.
Are CSV and PDF exports included?
Yes. After calculation, use the export buttons above the form. They save results, vertices, samples, and objective information.