2026-09-29 9 min

System of Inequalities Calculator

Free system of inequalities calculator: solve a1x + b1 ? 0 AND a2x + b2 ? 0 with per-row solution, intersection and graph.

Calculator

Two Inequality Rows

Row 1 slope a1

Row 1 constant b1

Row 2 slope a2

Row 2 constant b2

Row 1 operator

Row 2 operator

Results update live as you type

Status:1x + -2 >= 0 AND -1x + 6 <= 0[6, +inf)

Solution Summary

Row 11x + -2 >= 0 → [2, +inf)

Row 2-1x + 6 <= 0 → [6, +inf)

Intersection[6, +inf)

Results

Optimal

System solution (AND)

[6, +inf)

Points satisfying both rows

Row 1: 1x + -2 >= 0

[2, +inf)

First ray

Row 2: -1x + 6 <= 0

[6, +inf)

Second ray

Overlap?

Nonempty

Max of lower bounds, min of upper bounds

Results are recomputed live and verified against the step-by-step work below.

Visualizations

Boundary values

2Row1 bnd

6Row2 bnd

Feasibility

Feasible: [6, +inf)100%

Formula

Systems ask for points satisfying every row, so each row is solved alone and the rays are overlapped.

Row 1: a1x + b1 ? 0 AND Row 2: a2x + b2 ? 0 — answer is the overlap

Alternative Form

Each row gives a ray; the system keeps only points in both rays

Variables

Symbol	Name	Unit	Description
a1, b1	First row	numbers	Coefficients of a1x + b1 ? 0
a2, b2	Second row	numbers	Coefficients of a2x + b2 ? 0
ops	Operators	choices	One per row
overlap	Intersection	interval	Points common to both rows

Explanation

Solve each linear inequality as a ray, then intersect the rays: max of lower bounds, min of upper bounds.

Derivation

Row solutions are rays (-inf, r] or [r, +inf); their intersection is again an interval (possibly empty or a point).

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write both rows.

Row 1 and Row 2

r1:1x + -2 >= 0

r2:-1x + 6 <= 0

Result: System stated

Step 2

Solve Row 1 alone as a ray.

Row 1 ray

s1:[2, +inf)

Result: [2, +inf)

Step 3

Solve Row 2 alone as a ray.

Row 2 ray

s2:[6, +inf)

Result: [6, +inf)

Step 4

Intersect: lower bound is the max, upper bound is the min.

Overlap

iv:[6, +inf)

Result: [6, +inf)

Step 5

Test a point from the overlap in both original rows.

Verify

iv:[6, +inf)

Result: [6, +inf)

Final Answer

[6, +inf)

Worked Examples

See how the calculator works with different scenarios.

Scenario

Two opposing rays form a segment.

Inputs

a1:1

b1:-2

a2:1

b2:-6

op1:>=

op2:<=

Calculation

Row 1 [2,+inf), Row 2 (-inf,6], overlap [2,6]

Result

[2, 6]

Interpretation

Only points in both rays survive.

Understanding Your Result

What This Means

The headline result is the exact answer for the system of inequalities calculator inputs you entered, with supporting values shown beside it.

Why It Matters

Homework, exams and real-world models all reduce to this calculation; a verified numeric answer beats guessing.

Professional Interpretation

Read the headline result first, then scan the supporting rows and the step list to confirm the arithmetic before reusing the value.

Recommendations

•Verify the headline result against the steps
•Test one worked example by hand
•Check edge cases separately
•Keep full precision until the final display
•Restate the answer in the problem context

Educational Content

Introduction

This system of inequalities calculator evaluates your inputs with closed-form arithmetic and shows every intermediate value.

Concept Explanation

Solve each linear inequality as a ray, then intersect the rays: max of lower bounds, min of upper bounds.

Applications

▸Homework and exams

▸Engineering checks

▸Finance models

▸Science labs

▸Programming logic

▸Everyday estimation

Benefits

•Instant verified answer
•Step-by-step work shown
•Charts that expose mistakes
•Handles decimals directly
•Free with examples

Limitations

•Numeric inputs only
•Real-number results
•Display rounding to 6 decimals

Best Practices

•Enter numbers carefully
•Check the steps line by line
•Verify with one example
•Watch units in word problems
•State the final answer with context

Common Misconceptions

Myth: One input is enough.

Truth: Most problems need every coefficient; missing values change the answer.

Myth: Rounding early is fine.

Truth: Keep full precision until the final display to avoid drift.

Myth: The chart is decoration.

Truth: The chart is a second computation — mismatched bars mean a bad input.

Myth: Any rearrangement works.

Truth: Only equivalent operations preserve equality; follow the steps.

Practical Tips

•Double-check each input
•Read the steps in order
•Compare against one worked example
•Verify edge cases separately
•Keep units consistent
•Use Reset to restore defaults
•Copy the summary for notes

Common Mistakes

Typing a value in the wrong field

Match each label to the formula variables before calculating.

Dividing by zero

Check denominators and leading coefficients first.

Ignoring the steps

The step list shows exactly where a sign or arithmetic slip hides.

Rounding mid-calculation

Round only the displayed answer.

Forgetting to verify

Substitute the answer back into the original statement.

Assumptions

This calculator makes the following assumptions to provide estimates:

Real numbers: Inputs and outputs are real numbers.

Exact inputs: Coefficients are taken as given.

Standard notation: Conventional symbols and operator precedence.

Display rounding: Six decimals in outputs.

Limitations

Be aware of these limitations when using this calculator:

Numeric scope: Symbolic parameters need the advanced pages.

Single scenario: One input set per calculation.

No units conversion: Convert units before entering values.

Frequently Asked Questions

Answers to common questions about this calculator.

How do I solve a system of inequalities?

Solve each row separately, then keep only the overlap. For x >= 2 and x <= 6 the overlap is [2, 6].

What does AND mean here?

Every row must hold at once — unlike OR, a point failing any single row is out.

When is a system empty?

When the largest lower bound exceeds the smallest upper bound, the rays are disjoint.

Can the answer be one point?

Yes — when both boundaries coincide and both are inclusive, e.g. x >= 3 and x <= 3 gives x = 3.

Do strict operators matter?

They decide brackets at shared endpoints: x > 3 and x < 3 is empty, while >= with <= gives the point.

How do I verify?

Pick a number from the overlap and check it against every original row.

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

Showing your steps

Write one operation per line.

Checking answers

Substitute back every time.

Reading interval notation

Brackets include, parentheses exclude.

Word-problem setup

Name variables before computing.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Solution set: All values satisfying the given statement.
Interval notation: Compact bracket notation for ranges of the number line.
Critical value: A point where the expression is zero or undefined, splitting sign regions.
Verification: Substituting the answer back to confirm the original statement.
Closed form: An exact formula evaluated directly, not an approximation loop.
Edge case: Inputs where the usual rule breaks, such as zero denominators.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Math foundations Math is Fun — Index Purplemath — Lessons Wikipedia — Mathematics Cuemath — Math resources

Applicable Standards

Common Core HSA — creating and reasoning with equationsNCTM Algebra — representations and equivalence

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.