Use the calculator to check textbook exercises, classroom examples, and quick tutoring demos. Share links preserve the selected mode and coefficients so students and colleagues can revisit the same inequality setup.
How to choose the right inequality mode
Use this page when you want both the algebra and the picture: interval notation, a number line or shaded region, and the intermediate steps that explain why a point is included or excluded.
Pick the structure first
Use linear, compound, absolute value, quadratic, or rational mode for one-variable work. Switch to half-plane or system mode when you need the 2D shaded solution set instead of a number-line answer.
Watch strict and non-strict operators
Open endpoints and dashed boundaries correspond to strict inequalities. Closed endpoints and solid boundaries correspond to non-strict inequalities. If the operator changes, the visual output should change too.
Check domain restrictions in rational mode
A factor can cancel algebraically while still leaving a forbidden point from the original denominator. Read the final set together with excluded points so a hole is not mistaken for part of the solution.
Related tools
How to read the visual output
One variable modes
For 1D problems, the number line is the picture of the interval notation. Use it to see unions, excluded points, and whether a boundary is open or closed.
Two variable modes
For half-planes and systems, the shaded region is the set of points that satisfies every selected condition. Dashed lines exclude the boundary and solid lines include it.
Step log
The step list records sign flips, factor splits, and domain restrictions. If the answer surprises you, look for the step where the operator changed or where a denominator forced a point to stay excluded.
Results
Number line
Plane view
Teacher notes
FAQ
Which inequality types does this solver support?
The calculator covers linear, compound, absolute value, quadratic, and rational one-variable inequalities, plus half-planes and systems in two variables. Each mode explains every algebraic step and visualises intervals or polygons.
How are strict inequalities shown on the graphs?
Open endpoints and dashed boundaries appear automatically when you choose < or >. Non-strict operators close the endpoint and switch boundaries to solid lines so you can distinguish included points at a glance.
Which mode should I start with?
Start with the simplest structure that matches the problem: linear for ax+b, compound for chained bounds, absolute value for |ax+b|, quadratic for degree 2, and rational when a denominator matters.
Why is one point excluded even though the expression simplifies?
If the original expression had that point in the denominator, it stays excluded from the solution set even if a factor cancels later. Simplifying helps with sign analysis, but the domain restriction still comes from the original inequality.
What does the shaded side mean in 2D?
The shaded side is the set of points that satisfies the inequality. A dashed line means the boundary itself is excluded, while a solid line means points on the boundary are included.