Calculator inputs
y > 2x + 1, 3x - 2y <= 6, x^2 + y^2 < 25, and -2 < x <= 5.Graph and solution
Step-by-step explanation
Calculation steps will appear here.
Verification tables
Point tests, intersections, and boundary values will appear here.
Formula used
For a linear inequality written as Ax + By + C ◇ 0, the boundary is Ax + By + C = 0. A strict symbol creates a dashed line, while an inclusive symbol creates a solid line.
How to use this calculator
- Choose whether you want one inequality, a system, a one-variable solution, or a point test.
- Enter each inequality using x, y, numbers, parentheses, powers, and a valid comparison symbol.
- Set the graph range, boundary thickness, shading opacity, grid preference, and precision.
- Select Calculate and graph to view boundaries, shaded regions, steps, and verification tables.
- Use the copy, PNG, CSV, and print buttons to save or share the result.
Example data
| Example | Input | Expected graph behavior |
|---|---|---|
| Linear inequality | y > 2x + 1 | Dashed line with shading above it. |
| Inclusive boundary | 3x - 2y <= 6 | Solid line and the matching half-plane. |
| System | y >= x - 2y < 4 | Overlap of both shaded regions. |
| Nonlinear region | x^2 + y^2 < 25 | Interior of a dashed circle. |
| Compound inequality | -2 < x <= 5 | Number-line interval open at −2 and closed at 5. |
Understanding inequality graphs
Graphing an inequality quickly turns an algebraic statement into a visible region. The boundary clearly shows where both sides have equal values. Shading identifies every coordinate satisfying the original comparison exactly.
A strict algebraic inequality uses either the less-than or greater-than symbol. Its boundary is excluded, so the calculator draws dashes. Inclusive inequalities include equality and therefore use solid boundaries.
Linear inequalities usually divide the coordinate plane into two half-planes. A test point reveals which side satisfies the comparison. The calculator evaluates points to shade that region accurately.
Systems require every inequality to remain true at once. Their common shaded area is called the feasible region. This region may be bounded, unbounded, empty, or disconnected.
Nonlinear inequalities can create circles, curves, and irregular regions. The calculator samples the viewing window and traces approximate boundaries. Increasing the range can reveal missing portions of curves.
One-variable inequalities are displayed using interval and set-builder notation. Open endpoints represent strict comparisons, while closed endpoints include equality. Compound statements become two comparisons sharing the same variable.
Intersection points are especially useful in optimization and coordinate geometry. Linear boundaries produce exact intersections when their coefficients are detected. Values are rounded using the selected precision setting carefully.
Point testing provides a direct way to verify any coordinate. Each inequality is evaluated separately before the complete system is checked. A point belongs only when every required comparison succeeds.
Graph settings help match classroom exercises and practical models. You can change axis limits, opacity, line thickness, and theme. Zoom controls provide faster inspection without retyping the inequalities.
Exports make results easier to reuse in assignments and reports. PNG saves the graph, while CSV stores calculated values. Printing creates documents users can save directly as PDF.
Boundary tables provide numeric points for checking the plotted curve. These values reveal intercepts, symmetry, and changing direction. Comparing tables with graphs strengthens algebraic understanding and confidence.
Precision controls how displayed coordinates appear without changing internal calculations. More decimals help when boundaries intersect near awkward values. Fewer decimals create cleaner results for introductory classroom exercises.
An empty region means no point satisfies all entered conditions. This results from contradictory restrictions or separated shaded areas. Reviewing each condition separately usually exposes the conflicting requirement.
Unbounded regions continue indefinitely in at least one coordinate direction. They commonly appear when constraints limit only selected sides. Expanding the graph window helps confirm their continuing behavior.
Frequently asked questions
1. What does a dashed boundary mean?
A dashed boundary means equality is excluded because the inequality uses < or >.
2. What does a solid boundary mean?
A solid boundary means equality is included because the inequality uses ≤ or ≥.
3. Can I graph several inequalities?
Yes. Add rows and use system mode to view every region and their common overlap.
4. Can the calculator graph circles?
Yes. Expressions such as x^2 + y^2 < 25 create circular regions.
5. Does it support compound inequalities?
Yes. A chained statement such as -2 < x <= 5 is split into two conditions.
6. Why is an intersection missing?
The boundaries may be parallel, nonlinear, outside the visible window, or numerically difficult to detect.
7. What is a feasible region?
It is the set of points satisfying every inequality in the system simultaneously.
8. Can I test a specific coordinate?
Yes. Select point-test mode and enter the x-coordinate and y-coordinate.
9. Can I save the result?
Yes. Copy the summary, download the graph, export CSV data, or print to PDF.