Inequality Region Grapher
Choose a boundary curve y = f(x) and a relation (>, ≥, <, ≤). The calculator plots the boundary — dashed for strict inequalities, solid for inclusive — and shades the half-plane the inequality describes.
Two conventions made visual
An equation in two variables describes a curve; an inequality describes a region. Replace the equals sign in y = f(x) with a greater-than and the solution stops being a line of points and becomes everything on one side of it.
This tool draws that picture. It plots the boundary y = f(x), shades the side the inequality selects, and draws the boundary itself dashed or solid according to whether its own points belong to the solution.
Solving an inequality of this shape needs two decisions, both conventions to be remembered rather than derived: which side to shade, and whether the boundary counts. Neither is hard, and both are easy to get backwards with nothing to check against.
Drawing them removes the ambiguity. The shaded area answers the first and the line style the second, so a hand sketch can be compared against the picture rather than against a written rule.
How to use this calculator
- Enter the boundary function Any expression in x — a line, a parabola, a trigonometric curve. This is the f(x) that y is compared against, not the inequality itself.
- Choose the relation Greater than, greater than or equal, less than, or less than or equal. The choice sets both which side is shaded and how the boundary is drawn.
- Set the x range Two comma-separated numbers. The region extends beyond the window; what is drawn is the part inside the interval you asked for.
- Read the line style as part of the answer Dashed means the boundary is excluded, solid means it is included. That distinction is the entire difference between a strict inequality and an inclusive one.
How the region is drawn
The boundary function is compiled and sampled at 400 evenly spaced points, and the vertical extent comes from a percentile band of those samples rather than their outright minimum and maximum. That stops one extreme value squashing the curve flat, and it fixes the top and bottom edges the shading is drawn against.
The shaded area is a single closed shape. For a greater-than relation its outline runs along the top edge of the picture and back along the boundary curve; for a less-than relation, along the bottom edge instead. So the region is bounded above or below by the frame rather than by anything mathematical.
The boundary line is drawn from the same samples, broken wherever the function is undefined or jumps by an outlier amount, so a curve with a vertical asymptote is not joined across it. Whether that line is dashed or solid is decided purely by the relation: the two strict relations dash it, the two inclusive ones leave it solid.
Nothing is tested point by point. The relation alone decides which side is shaded, which is why the answer cannot be confused by a curve that wanders.
What each input means
- f(x) Boundary function — form field “Boundary f(x)”
- The curve separating the two half-regions. It is the right-hand side of the inequality, with y always on the left.
- rel Relation — form field “Relation”
- One of four: strict or inclusive, above or below. It controls the shading direction and the line style together.
- [xMin, xMax] x range — form field “x range”
- The horizontal window. It limits what is drawn, not what satisfies the inequality.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
Above a parabola
Shade y > x² − 4 from −4 to 4. The boundary dips below the axis in the middle and rises steeply at both ends, so the shaded area has a distinctly curved lower edge.
Inputs Boundary f(x) = x^2 - 4, Relation = gt, x range = -4, 4
- Inequality y > x^2 - 4
- Range x ∈ [-4, 4]
- Boundary Dashed — points on the curve are not included.
- Shaded region All points above the boundary.
Result y > x^2 - 4
The region is everything above the parabola, including the interior of its bowl. Points inside a parabola satisfy the greater-than relation just as points far above it do, which is easier to accept from the picture than from the algebra.
The boundary is dashed. A point exactly on the curve gives equality, and the strict relation excludes it — so the parabola itself is drawn but does not belong to the solution.
An inclusive relation against a wave
Shade y ≥ sin(x) from −6.28 to 6.28. The relation includes equality, and the boundary oscillates rather than curving one way.
Inputs Boundary f(x) = sin(x), Relation = ge, x range = -6.28, 6.28
- Inequality y ≥ sin(x)
- Range x ∈ [-6.28, 6.28]
- Boundary Solid — points on the curve are included.
- Shaded region All points above the boundary.
Result y ≥ sin(x)
The boundary is solid rather than dashed, because equality is included. Every point of the sine curve belongs to the solution, and the unbroken line says so.
The region's lower edge follows the wave while its upper edge is the flat top of the frame — a clear contrast between the boundary that is mathematical and the one that is only the edge of the drawing.
Below a straight line
Shade y < 2x + 1 from −3 to 3. A linear boundary gives the classic half-plane, which is the case most inequality exercises start from.
Inputs Boundary f(x) = 2*x + 1, Relation = lt, x range = -3, 3
- Inequality y < 2*x + 1
- Range x ∈ [-3, 3]
- Boundary Dashed — points on the curve are not included.
- Shaded region All points below the boundary.
Result y < 2*x + 1
The shading now runs downwards from the line to the bottom of the frame, the mirror image of the previous example. Reversing the relation reverses the side and nothing else.
A quick check on any half-plane is to test the origin. Here zero is less than one, so the origin satisfies the inequality and must fall inside the shaded area — which it does.
Reading the result
The frame is not a boundary
Only one edge of the shaded area means anything: the curve. The straight edges at the top or bottom and at the two sides are where the picture ends, and the true region continues past all three indefinitely.
Dashed and solid carry the whole distinction
The shaded areas for a strict and an inclusive version of the same inequality are identical to the eye. The only difference is the line style, which is why it is worth reading deliberately rather than glancing past.
When you would use this
Checking a hand-shaded sketch
Inequality questions are usually marked on the shading and the line style. Comparing your sketch against the drawn region checks both at once, which is quicker than re-deriving the rule.
Seeing one constraint of a larger system
A feasible region is an intersection of several inequalities. Plotting each separately shows what every constraint contributes before the overlap is worked out, which is a useful way to find a constraint that does nothing.
Assumptions and limitations
What this calculator assumes
- The inequality has y alone on the left and a function of x on the right.
- The boundary function is real-valued and defined across most of the chosen interval.
- The shading direction follows from the relation alone; no sample point is tested against the inequality.
- The vertical extent of the drawing comes from a percentile band of the sampled values, not from their full range.
Where it stops being the right tool
- One inequality at a time: overlapping several to find a feasible region is not supported.
- The form is fixed as y against f(x), so a genuinely two-variable relation such as a circle's interior cannot be entered.
- The region is clipped to the drawn window, and nothing marks where it has been cut off.
Common mistakes
Shading the wrong side after rearranging
Why it happens. Getting an inequality into the y-against-f(x) form often means dividing by a negative number, which reverses the relation. The rearranged expression looks right, and the flipped sign is easy to forget.
How to avoid it. Test one convenient point in the original inequality — the origin, if it is not on the boundary. If it satisfies the inequality it must lie in the shaded area.
Drawing a solid boundary for a strict inequality
Why it happens. A boundary is drawn before the shading and feels like part of the setup rather than part of the answer, so the line style gets decided by habit.
How to avoid it. Read the relation symbol before drawing. Strict relations dash, inclusive relations do not, and the tool applies exactly that rule.
Frequently asked questions
What does a dashed boundary mean?
That the inequality is strict, so points on the curve are not part of the solution. A solid boundary marks an inclusive relation, where they are. The shaded area itself is the same in both cases.
How is the shaded side decided?
From the relation alone. A greater-than relation shades everything above the curve and a less-than relation everything below, with above and below measured vertically at each x. No sample point is tested.
Can the inequality contain terms in y?
No. The form handled here is y compared against a function of x. A genuine two-variable relation such as x² + y² ≤ 1 describes a region no vertical comparison can express, and needs an implicit-curve solver.
Why does the shaded region stop at the edge of the picture?
Because the region is unbounded and the drawing is not. The straight edges at the top or bottom and at the sides are the limits of the window; the true solution set continues past them without end.