2026-09-29 9 min

Rational Equation Calculator

Solve a/(x + b) = c with the excluded value x ≠ −b flagged and the solution verified. Free rational equation solver.

Calculator

Rational Equation Inputs

Numerator a

Shift b Help

Excluded x = −b

Value c Help

Right side (nonzero)

Results update live as you type

Status:12/(x + 2) = 3x = 2x ≠ -2

Rational Summary

Equation12/(x + 2) = 3

Excludedx ≠ -2

Verdictx = 2 — Denominator nonzero; fraction reproduces c ✓

Results

Optimal

Solution x

From clearing the denominator

Excluded Value

x ≠ -2

Zeroes the denominator

Verdict

Genuine solution

Denominator nonzero; fraction reproduces c ✓

Fraction Check

3 (target 3)

Original fraction at x

Denominator clearing is exact; the solution is tested against the excluded value and the original fraction within 1e-9.

Visualizations

Numerator vs Result Scale

12a

2|x|

Solution Safety

Safe (≠ excluded)100%

Formula

Multiplying by the denominator trades a fraction equation for a linear one — provided the multiplier is never zero, which the exclusion enforces.

a/(x + b) = c → x = a/c − b, with x ≠ −b and c ≠ 0

Alternative Form

Cleared form: a = c(x + b), linear in x

Variables

Symbol	Name	Unit	Description
a	Numerator	number	Must be nonzero for c ≠ 0
b	Shift	number	Sets the excluded value −b
c	Target value	number	Must be nonzero
x	Unknown	number	Forbidden from equaling −b

Explanation

Rational equations hide a linear equation behind a division; the exclusion is the price of multiplying by a variable expression.

Derivation

a/(x+b) = c, x ≠ −b → multiply by (x+b): a = c(x+b) → x + b = a/c → x = a/c − b. Reject if x = −b.

Step-by-Step Calculation

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

Step 1

STEP 0 — name the excluded value that zeroes the denominator.

x + b ≠ 0

excluded:x ≠ -2

Intermediate: x + 2 = 0 → x = -2

Result: Forbidden: x = -2

Step 2

Multiply both sides by (x + b) to clear the denominator.

a = c(x + b)

cleared:12 = 3(x + 2)

Result: Denominator cleared

Step 3

Solve the linear remainder for x.

x = a/c − b

x:2

Intermediate: 12/3 − 2

Result: Candidate x = 2

Step 4

Check against the exclusion and the original fraction.

x ≠ excluded and a/(x+b) = c?

frac:3

Result: Denominator nonzero; fraction reproduces c ✓

Final Answer

Worked Examples

See how the calculator works with different scenarios.

Scenario

Clean denominator clearing.

Inputs

a:12

b:2

c:3

Calculation

x = 12/3 − 2 = 2; check 12/4 = 3 ✓

Result

x = 2 (excluded −2 avoided)

Interpretation

Solution 2 is safely away from −2.

Understanding Your Result

Your Result: a/(x + b) = c solved with exclusion enforced and fraction proof

Typical range: c ≠ 0 and a ≠ 0 for genuine solutions; x = −b always forbidden.

What This Means

The accepted x is the unique admissible value making the fraction equal c exactly.

Why It Matters

Rates, work-sharing and concentration problems divide by variable quantities — exclusions keep the math physical.

Professional Interpretation

State the exclusion before any algebra: it frames the whole solution and catches the classic rejected-candidate trap.

Recommendations

•Name x ≠ −b first
•Clear by multiplying whole sides
•Solve the linear rest
•Test the candidate vs exclusion
•Prove with the original fraction

Educational Content

Introduction

Rational equations divide by expressions containing x; clearing denominators reveals the linear core.

Concept Explanation

Multiply both sides by (x + b) — legal exactly when it is nonzero. The exclusion x ≠ −b preserves equivalence.

Applications

▸Work-rate sharing

▸Average speed problems

▸Concentration mixtures

▸Unit-cost splitting

▸Harmonic means

▸Lens equations

Benefits

•Exclusion flagged upfront
•Clearing shown explicitly
•Excluded-hit detection
•Original-fraction proof
•Impossible cases classified

Limitations

•Single fraction form a/(x+b) = c
•Numeric values
•Real solutions

Best Practices

•Write the exclusion first
•Multiply entire sides
•Solve linearly
•Compare candidate to exclusion
•Substitute into the fraction

Common Misconceptions

Myth: Multiplying by (x + b) is always safe.

Truth: Only with x ≠ −b — the exclusion is the license.

Myth: Excluded values can still solve.

Truth: They make the original undefined; they are never solutions.

Myth: c = 0 works like any target.

Truth: Nonzero/a can never equal 0 — impossible unless degenerate.

Myth: Clearing changes the solution set.

Truth: With the exclusion enforced, clearing is a true equivalence.

Practical Tips

•Exclusion before algebra
•Multiply whole sides
•Solve the linear core
•Compare with forbidden value
•Prove via original fraction
•Classify c = 0 directly
•Keep denominators visible

Common Mistakes

Forgetting the exclusion

x = −b must be named before solving.

Accepting an excluded candidate

Forbidden means forbidden — reject it.

Partial multiplication

Multiply every term by (x + b).

Allowing c = 0 loosely

Nonzero fractions never hit 0.

Skipping the fraction proof

Original-form check is the verdict.

Assumptions

This calculator makes the following assumptions to provide estimates:

Single fraction: Equation is a/(x+b) = c.

Real numbers: All values real.

Nonzero multiplier: Clearing licensed by exclusion.

Exact clearing: Doubles internally.

One unknown: Only x unknown.

Limitations

Be aware of these limitations when using this calculator:

One fraction form: Sums of fractions need combining first.

Numeric: No parameters.

Real domain: Complex poles excluded.

No inequalities: Equalities only.

Frequently Asked Questions

Answers to common questions about this calculator.

How do I solve 12/(x + 2) = 3?

Exclude x ≠ −2, multiply by (x + 2): 12 = 3(x + 2), so x = 2. Check 12/4 = 3 ✓.

What is an excluded value?

An x making any denominator zero — here x = −b. It can never be a solution.

Why multiply by the denominator?

To clear the fraction: both sides times (x + b) leaves a plain linear equation.

What if my answer equals the excluded value?

Reject it — the candidate makes the original undefined. Report no (admissible) solution.

Can c be zero?

Only degenerately: a nonzero numerator over a finite denominator is never 0.

How do I check rational solutions?

Substitute into the original fraction and confirm it equals c with a nonzero denominator.

Where do rational equations appear?

Work sharing, average speeds, mixtures and any per-unit split over a variable group.

Are extraneous roots possible here?

Yes — exactly when the candidate hits the exclusion. The check exists to catch it.

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

Exclusion first

License to multiply.

Clearing denominators

Whole sides, once.

Rejected candidates

Forbidden stays forbidden.

Work-rate stories

Sharing by division.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Rational equation: Equation with x in a denominator.
Excluded value: x zeroing a denominator (here −b).
Clearing denominators: Multiplying out fraction denominators.
Forbidden point: Excluded value on the number line.
Excluded-hit: Candidate equaling the exclusion.
Admissible solution: Candidate surviving exclusion + proof.
Degenerate: a = c = 0 edge.
Impossible: No admissible x exists.
Fraction proof: Original-form substitution.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Rational equations Math is Fun — Rational equations Purplemath — Solving rational equations Wikipedia — Extraneous and missing solutions Cuemath — Rational equations BYJUs — Rational equations

Applicable Standards

Common Core HSA-REI.A — rational equations with restrictionNCTM Algebra — domain awareness

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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.