≤∩≥System of Inequalities Calculator

2D linear systems, multi-constraint intersections, nonlinear & absolute value systems, linear programming, quadratic systems, feasibility testing, and compound inequalities — with step-by-step solutions.

Linear 2D: ax + by OP c (rows separated by |)
Format: a,b,op,c for ax+by OP c. Rows separated by |. op: <= ≤, >= ≥, <, >. Finds corner points and feasible region.
Steps: Full working
Quick Inserts
Classic Examples
2Dx+y≤4, x≥0, y≥02Dtriangle region2D4-constraint boxMulti−1 ≤ x ≤ 3Multi2x+1 ≤ 7, x/2 ≥ 1MulticontradictoryNonlinearinside circle, above lineNonlinearbetween parabolasNonlinearellipse ∩ half-plane|f| sys|2x-3|<5 and |x+1|≥2|f| sys|x|<3 and |x-1|>1LPmax 3x+2y, 3 constrLPmin x+y (cost)LPmax 5x+4y classicQuad sysx²-5x+4 and x²-1Quad systwo parabolasFeasibility(2,3) feasible?Feasibility(8,2) not feasibleCompound−3 < 2x−1 < 5Compound1 ≤ x²+1 ≤ 10

Key Theorems & Rules
Feasible Region \(\mathcal{F}=\bigcap_{i=1}^n H_i\)
Corner-Point Thm \(z^*\text{ occurs at a vertex}\)
Flip Rule \(a<0:\;ax\leq c\Rightarrow x\geq c/a\)
AND intersection \(S_1\cap S_2\cap\cdots\cap S_n\)
OR union \(S_1\cup S_2\cup\cdots\cup S_n\)
|f| \(\Leftrightarrow -k
|f|>k \(\Leftrightarrow f<-k\text{ or }f>k\)
Compound AND \(aa\cap f
2D corner \(\text{Solve pairs of boundary lines}\)
Test point \(\text{Check (0,0) in each ineq}\)
No solution \(l_i>u_j\text{ for some }i,j\Rightarrow\emptyset\)
Unbounded LP \(\text{No finite optimal value}\)
Shadow price \(\lambda_i=\partial z^*/\partial b_i\)
IVT \(f(a)f(b)<0\Rightarrow\exists\text{ root}\)
Sign chart \(\text{Test midpoint of each interval}\)
Input Format Guide
  • 2D / LP: a,b,op,c per row (ax+by OP c)
  • Multi: a,op,c per row (ax OP c)
  • Nonlinear: lhs,op,rhs (use x,y)
  • Abs Value: expr,op,k for |expr| OP k
  • Quadratic: a,b,c,op for ax²+bx+c OP 0
  • Feasibility: lhs,op,rhs rows + test (x,y)
  • Rows always separated by |
  • Operators: <= ≤, >= ≥, <, >
  • Variables: x (1D) or x, y (2D)
  • Powers: x^2, multiply: 2*x
Steps Modes
None  Answer only.
Summary  Methods as pills.
Detailed  Accordion: boundary lines, test points, corners, shading, sign chart, LP vertices.
Operations Guide
2D — Linear 2D
Half-planes, corner points, test-point shading, 2–4 constraints
∩ — Linear Multi
Single-var system: max lower bound, min upper bound → interval
f≤g — Nonlinear
Boundary curves + test points → feasible region in 2D
|f| — Abs Value Sys
Each |f(x)|k solved, then all intersected
LP — Linear Prog
Corner-Point Theorem: optimal at vertex; shadow prices
ax² — Quadratic Sys
Each ax²+bx+c OP 0 via Δ + parabola shape, then ∩
P∈? — Feasibility
Substitute (x₀,y₀) into every constraint; ✓/✗ per row
a
AND-split: f>lo AND f
System of Inequalities Calculator © 2026  · PHP Sign-Chart Engine + Bisection + Corner-Point LP + KaTeX

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