Solve and graph inequalities
Core method
- One-variable: isolate x in a·x + b (op) c. Subtract b, then divide by a. If a is negative, reverse the inequality sign.
- Compound: solve both parts and intersect the intervals.
- Two-variable: graph the boundary line a·x + b·y = c, then shade points where a·x + b·y (op) c holds.
- System: shade only points that satisfy every inequality (intersection of half-planes).
Quick steps
- Select a mode: single, compound, two-variable, or system.
- Enter coefficients and operators. Adjust x/y ranges for visibility.
- Press Solve & Graph to compute and draw results.
- Use the export buttons to save a CSV table or a PDF summary.
Sample inputs and expected outputs
| Type | Input | Expected |
|---|---|---|
| One-variable | 2·x + 3 ≤ 7 | x ≤ 2 |
| Compound | 1 < 2·x + 3 ≤ 7 | -1 < x ≤ 2 |
| Two-variable | x + y ≥ 4 | Half-plane above the line y = 4 − x |
| System | x + y ≤ 6 ∧ x − y ≥ 0 ∧ y ≥ 0 | Feasible region is a triangular wedge in quadrant I |
Common questions
1) Why does the sign flip when dividing?
Multiplying or dividing by a negative reverses order on the number line. That reversal changes < to > and ≤ to ≥, and vice versa.
2) What do open and closed circles mean?
An open circle means the boundary is not included (< or >). A closed circle means the boundary is included (≤ or ≥).
3) Why does my graph look empty?
Your ranges may not include the solution region, or the system could be infeasible. Expand x/y bounds and try again.
4) Can this solve non-linear inequalities?
This version focuses on linear forms for reliable plotting. You can approximate other forms by sampling, but exact symbolic results are not guaranteed.
5) What is the boundary line in two variables?
It’s the equality case a·x + b·y = c. The inequality shades one side of that line depending on which points satisfy it.
6) How is a system region computed here?
The canvas checks many grid points and shades those that satisfy every inequality. It’s a practical visualization of the intersection region.
7) Why do I see jagged shading edges?
Shading uses a step size for speed. Reduce the grid step for smoother edges, especially when zoomed in.
8) What do the downloads include?
CSV contains your inputs and the computed summary. PDF contains the same summary plus steps and notes for printing or sharing.