System of Inequalities Calculator
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The solution of a system of inequalities is the region where every inequality is true at once. For y ≥ 2x − 1, y < −x + 5 and x ≥ 0, the shaded overlap is a triangle with corners (0, −1), (2, 3) and (0, 5).
Green: points that satisfy every inequality. Dashed boundaries are not included.
| Region | bounded (a closed polygon) |
| Corner points | \(\left(0,\ -1\right)\), \(\left(2,\ 3\right)\), \(\left(0,\ 5\right)\) |
| Area of the region | 6 |
| Is (0, 0) a solution? | Yes |
Show steps
- Graph each boundary as an equation: solid line for ≤ or ≥ (points on the line count), dashed for < or > (they do not).
- For each inequality, test a point not on its line, such as (0, 0), and shade the side where it is true.
- The solution is where all the shaded regions overlap.
- Corner points (where boundary lines meet inside the region): \(\left(0,\ -1\right)\), \(\left(2,\ 3\right)\), \(\left(0,\ 5\right)\).
How to use the system of inequalities calculator
- Type one inequality per line using x and y.
- Press Graph. The overlap of all solution regions is shaded green.
- For straight-line boundaries you also get the corner points and, if the region is closed, its area.
Graphing by hand
- Graph each boundary as if it were an equation. Use a solid line for ≤ or ≥ and a dashed line for < or >.
- Pick a test point not on the line, often (0, 0). If it makes the inequality true, shade its side; otherwise shade the other side.
- The solution is where all shadings overlap.
Corner points and linear programming
In linear programming, the maximum or minimum of an objective like P = 3x + 2y over a closed region occurs at a corner point. For x + y ≤ 6, 2x + y ≤ 8, x ≥ 0, y ≥ 0, the corners are (0, 0), (4, 0), (2, 4) and (0, 6); P is largest at (2, 4), where P = 14.
Special cases
- No solution: the regions never overlap, as with y > x + 3 and y < x − 1 (parallel lines, opposite sides).
- Unbounded region: the overlap extends forever in some direction, so there may be no maximum.
Systems of inequalities practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
How do you solve a system of inequalities?
Graph each inequality, shading the side that makes it true, and find the region where all the shaded areas overlap.
When is the boundary line solid or dashed?
Solid for ≤ and ≥ (points on the line count), dashed for < and > (they do not).
What are the corner points?
The points where two boundary lines meet on the edge of the solution region. They matter in linear programming.
Can the boundaries be curves?
Yes. The calculator shades regions bounded by parabolas, circles and other curves; corner points are found for straight-line boundaries.
What if nothing is shaded?
The inequalities have no common solution in the view, which usually means the system has no solution.
Sources
- OpenStax College Algebra 2e: Systems of Nonlinear Equations and Inequalities
- OpenStax Intermediate Algebra 2e: Graphing Systems of Linear Inequalities
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