Understanding and graphing systems of inequalities can often be challenging, especially when dealing with multiple inequalities at once. Whether you’re a student learning algebra, a teacher preparing classroom materials, or a professional needing quick visualizations, the System of Inequalities Graph Calculator provides a fast and accurate way to graph inequality systems and identify feasible regions.

This online tool allows you to input up to three linear inequalities, define a custom viewport, and generate a detailed graph complete with shaded solution regions, boundary lines, and approximate area calculations. Instead of manually sketching graphs or spending time evaluating points, this calculator automates the entire process in seconds.

In this guide, we’ll walk through everything you need to know about the tool—how it works, how to use it effectively, example problems, features, tips, FAQs, and more.

What Is a System of Inequalities?

A system of inequalities is a set of two or more inequalities that are evaluated at the same time. The goal is to determine the feasible region, which represents all points that satisfy every inequality in the system.

Each inequality represents a half-plane on a coordinate graph. When multiple inequalities are combined, the overlapping region—if any—forms the solution.

For example:

• 2x+3y≤122x + 3y \le 122x+3y≤12
• x−y≥1x - y \ge 1x−y≥1

Graphing these manually requires evaluating boundary lines, determining shading direction, and identifying intersection areas. Our tool handles all of that visually and accurately.

Key Features of the System of Inequalities Graph Calculator

✔ Supports up to three inequalities
✔ Customizable viewport (Xmin, Xmax, Ymin, Ymax)
✔ Shaded feasible region
✔ Solid or dashed boundary lines based on inequality type
✔ Accurate pixel-based feasible point calculation
✔ Approximate area of the feasible region
✔ Instant visual output on a dynamic graph canvas
✔ Reset and calculate buttons for easy re-use

This makes the tool ideal for students, teachers, mathematicians, engineers, economists, and anyone working with linear programming or algebraic analysis.

How to Use the System of Inequalities Graph Calculator

Using the tool is simple and intuitive, even if you're new to graphing. Follow the steps below:

Step 1: Set the Viewport

At the top of the calculator, you’ll see four numerical inputs:

• Xmin
• Xmax
• Ymin
• Ymax

These values control what part of the coordinate plane you want to display.
For most problems, the default values (−10 to 10) work perfectly. But if your graph is too zoomed in or out, adjusting the viewport will help you capture the whole picture.

Step 2: Enter the Inequalities

Each inequality uses the standard linear form: ax+by  (≤,≥,<,>)  cax + by \; (\le, \ge, <, >) \; cax+by(≤,≥,<,>)c

You will see fields for:

• a
• b
• relation ( ≤ , ≥ , < , > )
• c

Fill in the values for each inequality.
Inequality 3 is optional—you can graph one, two, or all three inequalities.

Step 3: Click “Calculate”

Once the values are entered:

✔ The graph will display boundary lines
✔ The feasible region (if it exists) will be shaded
✔ The calculator will compute:

• Feasible points (approximate pixel count)
• Approximate area in square units

These details appear below the graph.

Step 4: Click “Reset” to Start Over

If you want to try a different system, simply click Reset, and the tool clears everything instantly.

Example: Graphing a System of Two Inequalities

Let’s walk through a complete example.

Given:

1. x+y≤6x + y \le 6x+y≤6
2. 2x−y≥32x - y \ge 32x−y≥3

Input Into the Calculator:

Inequality 1:

• a = 1
• b = 1
• relation = ≤
• c = 6

Inequality 2:

• a = 2
• b = -1
• relation = ≥
• c = 3

Leave the viewport at default values.

After clicking Calculate:

You will see:

• A solid boundary line for each inequality
• Shaded turquoise region showing all points satisfying both
• The intersection area
• Approximate feasible area (useful for optimization problems)

This makes it extremely easy to understand how the inequalities interact.

Why This Tool Is Useful

✔ For Students

Helps visualize algebra concepts, prepare homework, or check solutions instantly.

✔ For Teachers & Tutors

Create clear inequality graphs for lessons, worksheets, or demonstrations.

✔ For Linear Programming

Visualize constraints before solving optimization problems.

✔ For Professionals

Engineers, economists, and researchers can quickly test constraint models.

✔ For Visual Learners

The shaded region helps interpret inequality systems more effectively.

Tips for Accurate and Effective Graphing

1. Use integers when possible

This keeps boundary lines clean and easier to interpret.

2. Adjust the viewport if the region is off-screen

If results look empty, zoom out.

3. Remember boundary line rules

• ≤ or ≥ → solid line
• < or > → dashed line

The tool automatically follows this rule.

4. Try adding a third inequality

Sometimes the feasible region becomes a polygon or disappears—great for practice problems.

5. Explore real-world applications

Use inequality systems to model budgeting, production limits, physics boundaries, and more.

20 Frequently Asked Questions (FAQs)

1. What does the System of Inequalities Graph Calculator do?

It graphs one to three inequalities and shades the region that satisfies all of them.

2. What is a feasible region?

It’s the area where all inequalities in a system overlap.

3. Can this tool graph strict inequalities (< and >)?

Yes. It uses dashed lines for strict inequalities.

4. Can I change the graph scale?

Yes, by adjusting Xmin, Xmax, Ymin, and Ymax.

5. Do I need to enter all three inequalities?

No—one or two is enough. The third is optional.

6. What happens if the inequalities have no solution?

The tool will show an empty feasible region.

7. What if the region is too small or too large?

Adjust the viewport to zoom in or out.

8. Does the tool show intersection points?

It shows the region visually, but not coordinate intersections.

9. Can I use decimals?

Yes, decimal values for a, b, and c are supported.

10. Does the calculator handle vertical or horizontal lines?

Yes, including cases where a = 0 or b = 0.

11. Is the area calculation exact?

It is an approximation based on pixel sampling.

12. Can I use negative coefficients?

Absolutely—negative values are fully supported.

13. What happens if b = 0?

The inequality becomes a vertical line, which is handled correctly.

14. What devices does this tool work on?

It works on computers, tablets, and mobile browsers.

15. Can this tool help with linear programming?

Yes, it is excellent for visualizing constraints.

16. What if I make a mistake entering values?

Click Reset to clear all fields.

17. Does it graph nonlinear inequalities?

No, only linear inequalities in the form ax + by = c.

18. How do I know which shading represents the solution?

The tool uses turquoise shading to highlight feasible regions.

19. Can I screenshot the graph?

Yes—use your device’s screenshot feature.

20. Is this tool free?

Yes, it is completely free to use.

Final Thoughts

The System of Inequalities Graph Calculator is a powerful and user-friendly tool that makes learning and analyzing inequality systems easier than ever. Whether you're solving math assignments, exploring real-world constraints, or teaching algebra concepts, this tool provides an instant graphical representation of your inequalities along with valuable insights such as feasible points and area estimates.

Instead of manually drawing lines and shading half-planes, you can rely on this calculator to produce clean, accurate, and visually appealing graphs in seconds.