System of Inequalities Calculator
This system of inequalities calculator shades the region and, more usefully, gives you every corner point as an exact fraction along with the two lines that meet there.
Type x ≥ 0, y ≥ 0, x + y ≤ 4 and you get the corners (0, 0), (0, 4) and (4, 0).
What the System of Inequalities Calculator Gives You
Plenty of tools will shade the overlap for you, and some do it beautifully. What almost none of them give is the list of corner points — and that is nearly always the thing being asked for.
The reason is a small fact with large consequences: for anything linear, the biggest and smallest values over a closed region are always reached at a corner. So the corners turn a whole area you cannot search into four or five numbers you can.
Why Reading Them Off the Graph Fails
Corners are rarely whole numbers. 3x + 5y ≤ 47 together with 2x + 7y ≤ 41 meet at (124/11, 29/11).
On a grid that looks like about (11.3, 2.6), and any answer built on those two numbers is wrong from that point onwards. Each corner is where two lines cross, so it is the solution of a pair of simultaneous equations — and that solution is always an exact fraction.
Not Every Crossing Is a Corner
Two lines cross somewhere whether or not that place is anywhere near your region.
A crossing earns the name corner only if it also satisfies every inequality you did not use to find it. Checking that is mechanical and easy to skip, and skipping it is the second most common way to lose marks here.
Drawing It: Line, Test Point, Shade
Replace the inequality sign with an equals sign and draw that line. Then take any point that is not on it — the origin, if it is free — and put it into the original inequality. If it holds, shade that side.
A solid line means the points on it count. A dashed line means they do not, and that is exactly what decides whether a corner belongs to your answer. Which side to shade follows the same reasoning as any linear inequality, just applied to a line rather than a number.
When the Region Is Empty
x + y ≤ 1 with x + y ≥ 3 asks for a number that is at most 1 and at least 3.
The shaded halves never meet, so there is nothing at all. The picture comes out blank, which looks like a broken drawing and is not — the system genuinely has no solution, and saying so is the right answer. The same thing happens with a compound inequality on one line, where two conditions can also rule each other out.
When It Never Closes
A region can run off for ever in some direction. It still has corners, and a smallest value may still exist, but a largest one often does not.
That is worth checking before you spend time hunting for a maximum that is not there. It also explains an answer that looks incomplete: an open region is a normal outcome, not a missing constraint.
When It Flattens
Two constraints facing each other exactly — x ≥ 1 with x ≤ 1 — squeeze the region onto a line, or down to a single point.
There is no area left, so nothing gets shaded, and the drawing looks like it failed. It has not. A segment or a point is a perfectly good answer, and its ends are still corners.
Using the System of Inequalities Calculator
Type your inequalities separated by commas, semicolons or the word and. Up to eight are read, and both x and y are understood as they stand. You get each inequality turned the same way, the shape of the region, every corner as an exact fraction with the lines that made it, a mark on any corner a strict symbol leaves out, and a drawing with the strict boundaries dashed.
System of Inequalities Calculator FAQ
How do you solve a system of inequalities?
How do you find the corner points?
Is every crossing of two lines a corner?
What if there is no region at all?
Why do corners matter more than the shading?
Shade to see it; solve the pairs to know it.
