Did you know? The line that looks most harmless in a JC1 H2 Mathematics inequality solution is often the one that causes the mistake: cross-multiplying by an expression whose sign is unknown. In an equation, multiplying both sides by a non-zero denominator is routine. In an inequality, multiplying by a negative quantity reverses the inequality sign. If you do not know the denominator’s sign, you cannot simply keep the direction unchanged.
For JC1 H2 Mathematics learners in Bukit Timah, the Singapore-Cambridge A-Level H2 Mathematics 9758 syllabus includes inequalities involving rational expressions with linear or quadratic numerators and denominators. These questions are not solved safely by treating fractions as if they were equations. They require attention to zeros, undefined inputs, signs on intervals and whether endpoints belong in the final answer.
This JC1 H2 Mathematics Tutor Bukit Timah guide teaches one reliable method: the rational-inequality sign chart. You will see the decision rules, full worked examples, exclusions, quick checking questions and a tutor-style way to diagnose mistakes. It does not replace the broader Bukit Timah H2 Mathematics modulus and inequalities overview; it concentrates on the easily missed sign decision.
Quick answer: factor, mark, test and select
To solve an inequality such as (x − 2)/(x + 1) > 0, find the numbers where the numerator is zero or the denominator is zero. These are 2 and −1. Mark them on a number line, dividing it into three intervals. Test a value in each interval to find whether the expression is positive or negative. Select the intervals that match the inequality.
For this example, the fraction is positive for x < −1 and x > 2. The final answer is x < −1 or x > 2. We exclude −1 because the denominator is zero there, and we exclude 2 because the inequality is strict: it asks for greater than zero, not greater than or equal to zero.
- A numerator zero can be included for ≥ or ≤ if the original expression is defined at that point.
- A denominator zero is never included because division by zero is undefined.
- An inequality with > or < excludes the zero-value endpoints.
- Between consecutive zeros and poles, a factored rational expression keeps its sign unless another critical point intervenes.
- A sign chart tells you what is positive or negative; the question tells you which signs to keep.
Why cross-multiplying can give the wrong answer
Consider (x − 2)/(x + 1) > 0. A learner multiplies both sides by x + 1 and writes x − 2 > 0, concluding x > 2. That finds only one correct branch. The learner has silently assumed x + 1 is positive. But for x < −1, both numerator and denominator are negative, so the fraction is also positive. The missing branch comes from losing the denominator’s sign.
There are two legitimate ways to multiply without losing the logic. You could split the problem into cases x + 1 > 0 and x + 1 < 0, reversing the sign in the second case. Or you could use a sign chart, which organises these cases visually. The sign chart is often faster once the factors are clear.
Notice a deeper point. The inequality symbol is a comparison, not merely punctuation. Multiplying both compared numbers by a negative quantity changes which side is larger. If we know 3 > 1, multiplying by −2 produces −6 < −2. No amount of algebraic neatness can bypass that order property.
Worked example 1: one numerator zero and one denominator zero
Solve (x − 2)/(x + 1) > 0. The critical values are x = −1 and x = 2. These divide the real line into (−∞, −1), (−1, 2), and (2, ∞).
Test x = −2: (−4)/(−1) = 4, positive. Test x = 0: (−2)/(1) = −2, negative. Test x = 3: (1)/(4) = 1/4, positive. Therefore keep the first and third intervals.
The endpoint x = −1 is excluded regardless of the symbol, because it makes the expression undefined. The endpoint x = 2 makes the expression zero, which is not greater than zero. The solution set is (−∞, −1) ∪ (2, ∞).
A fast reasonableness check is to try a large positive and a large negative number. For both, the leading x terms make the ratio close to one, so both outer intervals should indeed be positive. This is a checking device, not a substitute for a complete sign chart.
Worked example 2: a quadratic numerator
Solve [(x − 3)(x + 2)]/(x − 1) ≤ 0. The numerator is zero at x = −2 and x = 3. The denominator is zero at x = 1. Order the critical values: −2, 1, 3.
Test x = −3: numerator (−6)(−1) is positive and denominator −4 is negative, so the expression is negative. Test x = 0: numerator (−3)(2) is negative and denominator −1 is negative, so the expression is positive. Test x = 2: numerator (−1)(4) is negative and denominator 1 is positive, so the expression is negative. Test x = 4: numerator (1)(6) and denominator 3 are positive, so the expression is positive.
We want negative or zero. Thus choose the interval before −2 and the interval between 1 and 3. Include −2 and 3 because they make the numerator zero and the denominator remains non-zero. Exclude 1 because the expression is undefined. The final answer is x ≤ −2 or 1 < x ≤ 3.
This example reveals two different endpoint decisions on the same question. At x = −2, a zero numerator is acceptable. At x = 1, no value exists. Marking circles on a number line without stating *why* they are open or closed can hide the most important reasoning.
Worked example 3: a repeated factor
Solve (x − 2)²/(x + 1) ≥ 0. The square in the numerator is non-negative for every real x and becomes zero only at x = 2. The denominator has a simple zero at x = −1.
For x < −1, the denominator is negative while the numerator is positive, making the fraction negative. For x > −1, the denominator is positive and the numerator is non-negative, making the fraction non-negative. Hence the solution is x > −1, with x = 2 automatically included.
Why does the sign not flip at x = 2? The squared factor touches zero but does not change from positive to negative. If students use the shortcut “alternate plus and minus signs at every marked point” without checking multiplicities, they can produce the wrong pattern. Test values remain the reliable method.
Worked example 4: a square in the denominator
Solve (x − 1)/(x + 2)² > 0. The denominator is positive for all x except x = −2, where it is undefined. Therefore away from that excluded point, the sign of the whole fraction is the sign of x − 1. The expression is positive exactly when x > 1.
The solution is x > 1. Even though −2 is an excluded input, it lies outside the chosen interval, so it does not need a separate exclusion in the final set. Still mark it when analysing the original function, so the domain logic is complete.
This example and the previous one show why repeated factors deserve separate attention. A zero of even multiplicity generally does not change the sign of that factor as we cross it, although a zero denominator remains forbidden.
What about cancellation and removable holes?
Suppose an expression is (x² − 1)/(x − 1). Algebraically it simplifies to x + 1 for x ≠ 1. But the original fraction is still undefined at x = 1. Cancelling the common factor does not grant permission to restore that input. Its graph follows the line y = x + 1 with a hole where x = 1.
If the question asks (x² − 1)/(x − 1) ≤ 0, the inequality becomes x + 1 ≤ 0, subject to x ≠ 1. The solution is x ≤ −1, which automatically avoids the hole. In a different inequality, the excluded input could lie inside the candidate range and would have to be removed explicitly.
The examination habit is simple: mark excluded values from the original denominator before any cancellation. Simplify only after recording the domain. Then apply the inequality to the simplified expression while preserving the original exclusions.
A four-line sign-chart worksheet
A sign chart need not be a large drawing with twenty arrows. Make a small table or number line with the critical values in increasing order. For each open interval, record the sign of each factor and multiply those signs. Test points are especially safe because they make fewer assumptions about repeated factors.
Write these four lines in order:
- Original inequality and domain restrictions.
- Factorised numerator and denominator, with critical values ordered.
- One sign for each interval, obtained by factoring or test values.
- Final interval notation, including or excluding zero endpoints as appropriate.
If the learner omits line one, they may include a pole. If they omit line two, they may overlook a zero. If they omit line three, the inequality direction may be guessed. If they omit line four, a correct analysis may not become a clear answer.
Five tiny diagnostic questions with solutions
Question 1. Is x = 2 included in (x − 2)/(x + 1) ≥ 0? Answer: Yes; the fraction is zero and 0 ≥ 0 is true. The denominator is 3, so the expression exists.
Question 2. Is x = −1 included in (x − 2)/(x + 1) ≤ 0? Answer: No; it is undefined because the denominator is zero, regardless of the inequality symbol.
Question 3. Solve (x + 3)/(x − 2) < 0. Answer: The critical values are −3 and 2. The fraction is negative between them, so −3 < x < 2. Neither endpoint is included.
Question 4. Solve (x − 4)²/(x − 1) < 0. Answer: The square is positive except at x = 4, and denominator negative when x < 1. Therefore x < 1. The zero at four lies outside the chosen interval.
Question 5. A student writes “multiply both sides by x − 5” in an inequality without conditions. What should be asked? Answer: Is x − 5 positive, negative or zero? Different signs create different cases, and x = 5 may be excluded from the original expression.
These are useful first checks before longer practice. A student who can answer the endpoints correctly but confuses the middle intervals may need more factor-sign practice. A student who cannot explain why denominator zeros are excluded needs a concept repair, not simply another sign chart.
Why a sketch and a sign chart should agree
A rational expression can be viewed as a graph of y = R(x). Solving R(x) > 0 means identifying where its graph lies above the x-axis. Solving R(x) ≤ 0 means where it lies on or below the axis, while respecting holes and vertical asymptotes. Your sign chart and your sketch are therefore two descriptions of the same solution.
The H2 syllabus permits use of approved graphing calculators for many equation and graph problems, but a calculator window may miss a narrow region or make a hole hard to see. The algebraic exclusions and sign reasoning remain essential. When a graph is used to solve an equation, follow the official requirements for sketches and working.
A practical check is to choose one sample point from each proposed solution interval and substitute it into the original inequality. Then choose one sample point from a rejected interval. The retained points should satisfy the comparison, and the rejected point should not.
What the right tutor repairs
A tutoring session should not begin with “all inequalities are easy if you draw a table.” It should begin with the student’s actual line of working. Did they cross-multiply too soon? Did they fail to factor the quadratic? Did they change a strict inequality into an inclusive one? Did they mistake a zero numerator for a forbidden point? Each error has a distinct cause.
A first-principles repair can begin with ordinary numbers: compare 4/2 with zero, then −4/2, then 4/−2 and −4/−2. Once the sign combinations are stable, replace numbers with factors. The learner discovers that a positive fraction needs matching signs, not necessarily positive numerator and positive denominator.
The next stage is a two-critical-point example, followed by a three-critical-point example and finally a repeated factor or removable hole. At each stage, the tutor asks the student to explain *which intervals* satisfy the original inequality and *why each endpoint* is included or excluded. This builds reasoning that survives new question forms.
Frequently asked questions
Can I always solve rational inequalities by drawing a graph?
Graphical reasoning is useful and may be allowed in H2 Mathematics questions, but you must still address excluded inputs, the correct intervals and the level of exactness requested. For rational expressions with clear factorisation, a sign chart often gives exact bounds most directly.
Does the inequality sign reverse when I cross-multiply?
Only when you multiply by a negative quantity. If the denominator contains x, its sign may depend on x, so you must split into cases or use an equivalent sign analysis. Blind multiplication is not safe.
Can a denominator root ever be included when the symbol is ≥?
No. An inequality sign cannot make division by zero defined. Domain exclusions always apply.
Why does a squared factor behave differently?
An even power is non-negative on both sides of its zero, so the factor generally does not switch sign as x crosses the root. The zero itself may still be a valid endpoint if it is in the numerator and the denominator is non-zero.
Is this an A-Level H2 Mathematics topic?
Yes. The 2027 syllabus 9758 lists solving rational inequalities with linear or quadratic expressions in the functions-and-graphs section. Schools may introduce it at different points of JC1 or JC2.
Keep exploring
The official 2027 H2 Mathematics syllabus 9758 specifies the tested strand. Explore the sibling guides on why f(x + a) shifts left, recovering uₙ from Sₙ, and tracing parametric curves to practise other JC1 foundations. The Bukit Timah H2 Mathematics topic overview gives the broader route.
The immutable eduKateSG small-group Maths tutorial reference explains the practice of locating the first incorrect step and rebuilding from there. Its specific details concern Secondary 1; JC1 support and availability should be confirmed through the Bukit Timah JC1 tuition page.
A good sign chart is not a trick for getting marks. It is a picture of where an expression can be trusted to be positive, negative, zero or not defined. Once those four states are clear, the final inequality becomes readable.