Algebra • Linear Systems

System of Linear Inequalities Point Checker

Verify whether an ordered pair (x₀, y₀) satisfies a system of simultaneous linear inequalities with step-by-step constraint checking and an interactive coordinate graph.

Reviewed by Sanjay Samanta & Academic Review Board

Last Updated: September 2026

Verified Accurate: Convex Optimization & Polyhedral Geometry

Interactive Calculator

System of Linear Inequalities Point Verifier

Presets:

System Inequalities (Ax + By [op] C)

Coordinates of Test Point (x₀, y₀)

Point x-coordinate (x₀)

Point y-coordinate (y₀)

Feasible Region Plot Range: [-10, 10]

Feasible Overlap Region

Point (x₀, y₀)

System Satisfaction

Satisfies All

Point lies inside the feasible intersection

Inequalities Passed

3 / 3 Passed

100% of constraints satisfied

Feasible Set Geometry

Convex Polyhedral Region

Simultaneous half-plane intersection

Constraint-by-Constraint Evaluation

Must Satisfy Every Condition

Constraint #	Inequality	Substitution (LHS)	LHS vs. RHS	Constraint Status

Step-by-Step Algebraic System Proof

Direct Answer & Overview

Verified Educational Guide

How to Check if a Point Satisfies a System of Linear Inequalities

An ordered pair (x₀, y₀) satisfies a system of linear inequalities if and only if it satisfies every single inequality in the system simultaneously. To verify, plug x₀ into x and y₀ into y for each inequality. If all resulting arithmetic statements are true, the point lies inside the overlapping feasible region. If the point fails even a single inequality, it is rejected and does not belong to the solution set.

Primary Mathematical Formula Simultaneous Half-Plane Intersection Test

Standard Equation

ƒ(x)

Q.E.D.

\begin{cases} A_1(x_0) + B_1(y_0) \le C_1 \\ A_2(x_0) + B_2(y_0) \le C_2 \\ \quad \vdots \\ A_k(x_0) + B_k(y_0) \le C_k \end{cases} \iff (x_0, y_0) \in \bigcap_{i=1}^k \mathcal{H}_i

All constraints must be satisfied simultaneously (Logical AND conjunction). Feasible region = H_1 ∩ H_2 ∩ ... ∩ H_k.

Exact Formula

Input Parameters

Required

System of Inequalities: Multiple linear inequalities expressed in standard or slope form.

Test Point (x₀, y₀): Coordinates of the candidate solution.

Expected Outputs

Calculated

System Verdict: True (Feasible / Satisfies all) vs. False (Infeasible / Violates one or more).

Constraint Scorecard: Table indicating individual pass/fail status for each inequality.

Feasible Region Geometry: Visualized intersection of half-planes on a Cartesian plot.

Worked Numerical Example

Instant Verification

Check if (1, 2) satisfies the system (x + y ≤ 4, y ≥ 0, x ≥ 0)

→ Check 1: 1 + 2 = 3 ≤ 4 (True). Check 2: 2 ≥ 0 (True). Check 3: 1 ≥ 0 (True). All 3 conditions pass.

True: Point (1, 2) lies within the triangular feasible region.

What Is a System of Linear Inequalities?

A system of linear inequalities consists of two or more linear inequalities involving the same set of variables (typically x and y). While a single linear inequality defines an infinite half-plane on one side of a boundary line, a system defines the simultaneous intersection of multiple half-planes.

Mathematically, if inequality 1 defines half-plane ℋ₁, inequality 2 defines ℋ₂, and so forth, the solution set 𝒮 of the entire system is the set-theoretic intersection:

𝒮 = ℋ₁ ∩ ℋ₂ ∩ … ∩ ℋ_k = { (x, y) ∈ ℝ² | ∀ i ∈ {1,…,k}, A_ix + B_iy [op] C_i }

This means that a point is an allowable solution if and only if it falls inside the overlapping zone shared by every single constraint in the system. Check individual lines using our single inequality point checker.

The Geometry of the Feasible Region

When multiple half-planes intersect on the 2D coordinate plane, they form a geometric region known as the feasible region. In convex geometry, this shape is always a convex polygon or polyhedral set:

Bounded Region

Enclosed entirely by boundary lines forming a closed polygon (e.g. triangle, quadrilateral). It has a finite area and a finite set of extreme corner vertices.

Unbounded Region

Extends infinitely in at least one direction (e.g. first quadrant x ≥ 0, y ≥ 0). It possesses corner vertices but has infinite total area.

Empty (Infeasible)

Occurs when constraints contradict one another (e.g. y ≥ x + 4 and y ≤ x - 1). No point exists that satisfies both lines, so 𝒮 = ∅.

The Logical Conjunction (AND) Principle

The defining algebraic rule of a mathematical system is that the conditions are joined by a logical conjunction (AND, ∧) rather than a disjunction (OR, ∨).

Condition 1: P satisfies Inequality 1 (True / False)

AND Condition 2: P satisfies Inequality 2 (True / False)

AND Condition 3: P satisfies Inequality 3 (True / False)

Overall Truth Value = (C₁ ∧ C₂ ∧ … ∧ C_k)

If even a single inequality evaluates to FALSE, the entire conjunction becomes false, and the point is disqualified from the solution set. Explore linear systems solvers with our system of equations calculator.

Real-World Applications: Linear Programming & Optimization

Systems of linear inequalities are not merely academic drills; they form the operational foundation of Linear Programming (LP), which drives modern logistics, industrial engineering, and financial portfolio management:

Manufacturing Resource Allocation

A factory produces tables (x) and chairs (y). Limited wood and labor hours create inequalities: 2x + y ≤ 100 (wood) and x + 3y ≤ 120 (labor). Testing a production plan like (30, 20) verifies whether it is physically feasible.

Nutritional & Dietary Constraints

Dietitians design meal plans subject to calorie, protein, and sodium bounds: x + y ≥ 50 (protein grams) and 200x + 150y ≤ 2000 (calories). Verifying a meal recipe ensures adherence to strict clinical health targets.

Step-by-Step Verification Algorithm

Extract the Candidate Coordinates: Designate the point as (x₀, y₀).

Substitute into Inequality 1: Compute the left-hand side arithmetic. If inequality 1 is false, you may terminate immediately: the point is not in the system's solution set.

Repeat for All Successive Inequalities: Continue sequentially through inequality 2, 3, etc., recording the truth value of each constraint.

Formulate the Conjunction Verdict: If all constraints are marked TRUE, declare the point a valid solution. If any constraint is FALSE, reject the point.

Step-by-Step Graded Worked Examples

Example 1 • 3-Constraint System Satisfied Basic Tier

Determine if P(2, 1) satisfies: { x + y ≤ 5, 2x - y ≥ 1, y ≥ 0 }

Constraint 1: (2) + (1) = 3 ≤ 5 ⇒ TRUE

Constraint 2: 2(2) - (1) = 4 - 1 = 3 ≥ 1 ⇒ TRUE

Constraint 3: (1) ≥ 0 ⇒ TRUE

Verdict: All 3 constraints are satisfied ⇒ P(2, 1) belongs to the feasible region.

Example 2 • Single-Constraint Failure Disqualification Intermediate Tier

Determine if Q(4, 2) satisfies: { x + 2y ≤ 10, 3x - y ≤ 8 }

Constraint 1: (4) + 2(2) = 4 + 4 = 8 ≤ 10 ⇒ TRUE

Constraint 2: 3(4) - (2) = 12 - 2 = 10 ≤ 8 ⇒ FALSE (10 is not ≤ 8)

Verdict: Fails Constraint 2 ⇒ Q(4, 2) is NOT a solution to the system.

Example 3 • Point on Intersection of Boundaries Advanced Tier

Verify if R(3, 2) satisfies: { 2x - 3y = 0 (solid), x + y < 6 (dashed) }

Constraint 1: 2(3) - 3(2) = 6 - 6 = 0 ≤ 0 ⇒ TRUE (lies on boundary)

Constraint 2: (3) + (2) = 5 < 6 ⇒ TRUE (strictly inside dashed boundary)

Verdict: Both constraints pass ⇒ R(3, 2) is a valid feasible solution.

Common Pitfalls and System Verification Errors

Stopping After the First True Result

Assuming that because a point satisfies Inequality 1, it satisfies the whole system. A point must pass every single inequality without exception.

Treating Dashed Boundary Intersections as Solutions

If two lines intersect, but one of the lines is dashed (< or >), that vertex point is excluded from the solution set because it does not satisfy the strict inequality.

Ignoring Non-Negativity Constraints

Real-world systems frequently include implicit non-negativity bounds (x ≥ 0, y ≥ 0). Neglecting these constraints leads to testing negative coordinate pairs that are physically meaningless.

Arithmetic Sign Reversal with Multiple Minuses

Evaluating expressions like x - 3y ≥ 4 with negative points like (2, -2): students mistakenly write 2 - 6 = -4 instead of 2 - 3(-2) = 2 + 6 = 8.

Fact-Checked & Verified • Computational Accuracy Standards

Updated September 2026 • Editorial Policy

Authored By

Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By

Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

Found an error or have an improvement suggestion? Report a calculation issue

Frequently Asked Questions

How do you determine if a point satisfies a system of linear inequalities?

To determine if an ordered pair (x₀, y₀) satisfies a system of linear inequalities, substitute the point into every inequality in the system independently. If the point makes every single inequality true simultaneously, it satisfies the system and lies within the feasible region. If it violates even one inequality, it is not a solution.

What is a feasible region in a system of linear inequalities?

The feasible region (or solution set) is the geometric intersection of all the individual shaded half-planes defined by the inequalities in the system. Any coordinate pair located inside this overlapping region satisfies all constraints simultaneously.

Can a feasible region be empty or unbounded?

Yes. An unbounded feasible region extends infinitely in one or more directions (e.g. x ≥ 0, y ≥ 0). An empty (infeasible) region occurs when the inequalities contradict each other (e.g. y > x + 3 and y < x - 2), meaning no point can satisfy both conditions.

How are boundary intersections handled when strict and non-strict inequalities meet?

If an intersection vertex lies on both a solid boundary line (≤ or ≥) and a dashed boundary line (< or >), the vertex point itself does NOT belong to the solution set because points on dashed lines are excluded from strict inequalities.

What is the connection between systems of inequalities and linear programming?

In operations research and linear programming, systems of linear inequalities define the constraints (budget, labor, raw materials) that enclose the feasible region. The fundamental theorem of linear programming proves that optimal values (maximum profit or minimum cost) always occur at the corner vertices of this region.

What is the fastest way to check if the origin satisfies a system?

Plug (0, 0) into each inequality. The variable terms drop out completely, leaving only 0 compared to the constant C. For example, in 3x + 4y ≤ 12, plugging in (0, 0) yields 0 ≤ 12, which is immediately confirmed true.