Understanding Linear Inequality Systems
A system of linear inequalities describes many limits at once. Each inequality draws a half plane. The shared part is the solution set. This calculator focuses on two variables. It reads each boundary line and tests the side selected by the sign.
Why Feasible Regions Matter
The feasible region shows every ordered pair that satisfies all rules. It may be a polygon, a ray shaped area, a strip, a single edge, or an empty set. In optimization, only feasible points are allowed. In algebra, the shaded overlap explains the answer more clearly than one line alone.
How the Method Works
The tool rewrites each statement into standard form. It then compares boundary lines in pairs. Where two nonparallel lines meet, a possible corner appears. That point is tested against every inequality. Passing points become feasible vertices. The same tests also check any point entered by the user.
Reading the Results
A bounded result lists corner points and coordinate ranges. An unbounded result may still have valid solutions, but it extends forever. If no test point can satisfy every rule, the system is likely inconsistent. Strict signs remove boundary points. The calculator notes this when it checks values.
Practical Uses
Students can compare hand graphs with exact vertices. Teachers can create answer keys. Operations teams can model capacity, budget, and demand limits. Designers can test safe ranges. The same ideas also support linear programming. When an objective is supplied, the best value is checked at feasible vertices when possible.
Good Input Habits
Write one inequality on each line. Use x and y as variables. Decimals, negatives, and simple fractions are accepted. Use <=, >=, <, or > for signs. Keep multiplication simple, such as 2x instead of 2*x. Review the parsed table before trusting the final answer.
Common Mistakes to Avoid
Many wrong answers come from shading the wrong side. Test a simple point after drawing each line. Watch negative signs when moving terms across the symbol. Parallel limits can create a strip, not a corner. Also remember that strict inequalities use open boundary lines. A point on that line is not included, even when it looks close. Round only after all tests finish. This keeps answers stable during review.