Secondary 4 G2 Mathematics Tutor Bukit Timah | Fractional Equations and Forbidden Values. Did you know? A perfectly good-looking algebraic answer can be impossible in the original question. Solve (x² − 9)/(x − 3) = 6 by cancelling x − 3 and you appear to obtain x = 3. Yet the original denominator becomes zero at x = 3. That means the equation has no permissible solution, not x = 3.
This is why Secondary 4 G2 Mathematics tuition in Bukit Timah should teach fractional equations as a reasoning process, not merely an exercise in clearing denominators. The student must identify excluded values, build equivalent steps carefully, solve the resulting equation and check every candidate in the original expression.
The official 2027 Singapore-Cambridge SEC G2 Mathematics K210 syllabus specifies simple fractional equations reducible to linear equations and equations reducible to quadratics under N7: Equations and inequalities. The related N5 strand covers operations on algebraic fractions. A candidate graduating in 2026 should follow the applicable N(A)-Level Mathematics Syllabus A 4045 instead of the SEC K210 code introduced for 2027. Use SEAB’s 2026 syllabus directory or the 2027 G2 directory for the right year.
Quick answer: four steps for every fractional equation
- State excluded values first: solve each original denominator = 0 and record values that x cannot take.
- Clear denominators legally: multiply the whole equation by a common denominator, understanding that it is nonzero for permitted x.
- Solve the resulting equation: use linear methods or quadratic methods as appropriate.
- Check each result against the original: reject forbidden values and confirm that the original two sides match.
What does a forbidden value mean?
A fraction with denominator zero is undefined in real-number arithmetic. For 3/(x − 2), x = 2 is forbidden. For 1/(x − 2) + 2/(x − 3), both x = 2 and x = 3 are forbidden. Writing those restrictions before simplifying prevents a cancellation from silently erasing the original equation’s domain.
| Original denominator | Forbidden value(s) | Why |
|---|---|---|
| x − 2 | x = 2 | Denominator becomes zero. |
| x + 4 | x = −4 | Denominator becomes zero. |
| x² − 9 | x = −3 or 3 | Factor as (x − 3)(x + 3). |
| x(x − 5) | x = 0 or 5 | Either factor can vanish. |
| (x − 1)² | x = 1 | A squared zero remains zero. |
Did you know? A cancelled factor still matters. The expressions (x² − 9)/(x − 3) and x + 3 have the same value at every permitted x, but only the first excludes x = 3. A tutor should teach this distinction explicitly before asking students to solve a longer equation.
Seventeen original worked examples, with checks
1. Clear an ordinary numerical denominator
Solve x/3 + (x − 2)/4 = 3. Multiply every term by 12: 4x + 3(x − 2) = 36; 7x − 6 = 36. Therefore x = 6. Check: 2 + 1 = 3.
2. One denominator containing x
Solve 3/(x − 2) = 6, where x ≠ 2. Multiply: 3 = 6(x − 2). Then x − 2 = 1/2, so x = 5/2.
3. A ratio of linear expressions
Solve (x + 1)/(x − 3) = 2, where x ≠ 3. Thus x + 1 = 2(x − 3), yielding x = 7. Check 8/4 = 2.
4. A simple reciprocal equation
Solve 5/(x + 4) = 1, where x ≠ −4. Multiply by x + 4: 5 = x + 4, so x = 1.
5. Fractions on both sides
Solve 2/(x − 1) = 3/(x + 1), x ≠ 1, −1. Cross-multiplying gives 2(x + 1) = 3(x − 1), so x = 5. Check 2/4 = 3/6.
6. Another two-denominator equation
Solve 1/(x − 2) = 2/(x + 1), x ≠ 2, −1. Then x + 1 = 2(x − 2), giving x = 5. Check: 1/3 = 2/6.
7. A fractional equation leading to a quadratic
Solve 6/(x + 4) = x + 3, x ≠ −4. Multiply: 6 = (x + 4)(x + 3), so x² + 7x + 6 = 0. Factor (x + 1)(x + 6) = 0. Solutions are x = −1 or −6, both permitted.
8. An extraneous value that must be rejected
Solve (x² − 9)/(x − 3) = 6, x ≠ 3. For allowed x, simplify to x + 3 = 6, suggesting x = 3. Since this violates the restriction, there is no solution.
9. Cancelling a denominator does not restore its zero
The equation (x − 2)/(x − 2) = 1 holds for all real x except 2, not for every real x. Cancelling gives 1 = 1 on the permitted domain.
10. Candidate equals a forbidden value
Solve x/(x − 2) = 2/(x − 2), x ≠ 2. Multiplying by x − 2 gives x = 2, which is forbidden. The equation has no solution.
11. Two sides can be incompatible
Solve 1/(x − 1) = 1/(x + 1), x ≠ 1, −1. Multiplication gives x + 1 = x − 1, impossible. Thus no real solution.
12. Solve an expression with a constant term
Solve 4/(x − 1) + 1 = 3, x ≠ 1. Subtract 1, so 4/(x − 1) = 2, giving x − 1 = 2 and x = 3.
13. Keep bracketed numerators together
Solve (2x + 1)/3 = (x − 4)/2. Multiply by 6: 2(2x + 1) = 3(x − 4), so 4x + 2 = 3x − 12 and x = −14. Both original sides equal −9.
14. Factor a quadratic denominator
Solve 1/(x² − 9) = 1/16, x ≠ −3, 3. Multiply to obtain x² − 9 = 16; x² = 25. Thus x = −5 or 5, both allowed.
15. A simple square denominator
Solve 4/(x − 1)² = 1, x ≠ 1. Multiply: (x − 1)² = 4. Then x − 1 = ±2, giving x = −1 or 3, both valid.
16. A fraction and a constant comparison
Solve (x + 5)/(x − 1) = 3, x ≠ 1. Multiply: x + 5 = 3x − 3, giving 8 = 2x and x = 4. Original check: 9/3 = 3.
17. A more demanding two-denominator quadratic
Solve 1/(x − 2) + 2/(x − 3) = 5, x ≠ 2, 3. Combine: (3x − 7)/[(x − 2)(x − 3)] = 5. Then 3x − 7 = 5x² − 25x + 30, so 5x² − 28x + 37 = 0. Quadratic formula gives x = (14 ± √11)/5. Neither value equals 2 or 3, so both are permissible.
Why solving and simplifying are two different tasks
Simplifying (x² − 9)/(x − 3) rewrites the expression as x + 3 with x ≠ 3. Solving an equation involving that expression means finding which permitted x-values satisfy a given equality. A student who copies “x + 3” as the answer to a solve question has stopped before solving; a student who forgets x ≠ 3 may return a forbidden solution.
Sometimes clearing denominators introduces algebraic candidates that the original problem cannot accept. These are often called extraneous solutions. The final substitution into the original equation catches them. It also detects ordinary arithmetic mistakes, making it a valuable habit even when no candidate is excluded.
How a tutor can diagnose six recurring errors
| Student’s first wrong line | Likely cause | Useful follow-up |
|---|---|---|
| Multiplies only one term by the common denominator | Equality not treated as a whole | Write every term on both sides before cancelling denominators. |
| Writes no restrictions before clearing x − 2 | Domain of the equation ignored | What value makes the original denominator zero? |
| Cancels (x − 3) and allows x = 3 afterward | Removed factor confused with removed restriction | Check x = 3 in the original expression. |
| Solves the derived quadratic but retains only one root | All candidate roots not examined | Use the zero-product rule or ± carefully. |
| Rejects a negative solution just because it is negative | Sign mistaken for invalidity | Does the original problem impose a positive-context restriction? |
| Finds a correct algebraic value but never substitutes back | Candidate not verified | Do the original left and right sides match? |
Five changed questions for a no-hint check
- Solve 2/(x − 4) = 1, stating the restriction.
- Solve 3/(x + 2) = x + 4, stating the restriction.
- Solve (x² − 4)/(x − 2) = 4, and check the resulting candidate.
- Solve 1/(x − 1) = 2/(x + 2), stating the restrictions.
- Describe exactly which values satisfy (x − 5)/(x − 5) = 1.
Answers: (1) x = 6, x ≠ 4; (2) 3 = (x + 2)(x + 4) gives x² + 6x + 5 = 0, so x = −1 or −5, with x ≠ −2; (3) simplification suggests x = 2, but x = 2 is forbidden, so no solution; (4) x + 2 = 2(x − 1), so x = 4, with x ≠ 1, −2; (5) all real x except 5. The restriction belongs to the original expression, not merely the final simplified one.
A useful 90-minute three-pupil tutorial model
| Minutes | Learning activity | Independent evidence |
|---|---|---|
| 0–10 | Recall denominator restrictions from earlier algebra | Learner identifies forbidden values without hints. |
| 10–25 | Unseen diagnostic with two linear fractional equations | Tutor identifies the first illegal transformation. |
| 25–50 | Contrast valid solutions with extraneous candidates | Student can explain why a cancelled factor still matters. |
| 50–75 | Individual new linear and quadratic fractional equations | Every candidate is checked in the original. |
| 75–90 | Exit problem and planned delayed retest | Learner catches a forbidden result several days later. |
The unchanged eduKateSG small-group Mathematics tutor reference describes premium three-pupil, 1.5-hour weekly tutorials near Sixth Avenue MRT. That educational baseline is used here to illustrate close diagnosis and feedback, not to promise an available Secondary 4 G2 class, current fees or a particular tutor.
A seven-day fractional-equation repair cycle
- Day 1: identify restrictions in ten rational expressions.
- Day 2: solve simple fractional equations reducible to linear equations.
- Day 3: clear multiple denominators with a common multiple.
- Day 4: solve two equations that produce quadratic equations.
- Day 5: analyse deliberately misleading extraneous solutions.
- Day 6: attempt an unseen mixed set and log the first wrong step.
- Day 7: retest the highest-frequency mistake with changed expressions.
Frequently asked questions
Why must x ≠ 3 survive after cancelling x − 3?
Because the original denominator becomes zero at x = 3. Algebraic cancellation shows equality on the common permitted domain; it does not redefine the original expression at a point where the expression did not exist.
Is this topic included in G2 SEC Mathematics?
Yes. The 2027 K210 N7 syllabus explicitly lists simple fractional equations reducible to linear equations and examples reducible to quadratic equations. The related algebraic-fractions techniques appear in N5. Use the actual cohort’s syllabus to select appropriate practice.
Is cross-multiplication always safe?
It is a shorthand for multiplying both sides of an equality between fractions by a common nonzero denominator. Before applying it, state denominator restrictions. If several fractions are added, a common-denominator method is often clearer than a quick cross-multiplication shortcut.
How can parents check that the method has been mastered?
Ask the student to solve a new rational equation, list prohibited values and substitute every proposed root into the original. Do this again after an interval so the explanation is independent, rather than freshly copied from the tutor.
Read more Secondary 4 G2 Mathematics topics
- Algebraic fractions and common denominators
- Equations and simultaneous solutions
- Quadratic formula and completing the square
- Changing the subject of formulae
Official references: 2026 SEAB N(A)-Level syllabuses · 2027 SEAB SEC G2 syllabuses · K210 N5 and N7 syllabus details. For the original premium teaching format, see the unchanged eduKateSG Mathematics tutorial reference or enquire via eduKate Singapore. Every algebra example is original educational practice, not an official examination question.