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Contents · 0% complete
  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn7367Paper 1 · Compulsory contentFurther algebra and functionsRational inequalities
Further algebra and functions · Subtopic 4 of 4

Rational Inequalities

Never multiply an inequality by an expression that might be negative. Multiply by the square of the denominator instead (always positive), collect on one side, factorise, and use a sketch or sign table. Exclude values that make a denominator zero.

Edexcel YFM01 WFM02Edexcel 9FM0 OptionAQA 7367 Paper 1AQA 7367 Paper 2≈ 35 minRevision 1 · updated 2026-10-27
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Section 1 of 14

Learning objectives

  • Explain why you multiply by the square of the denominator.
  • Find critical values and solve rational inequalities.
  • State solutions excluding asymptote values.
Pause and think

Solve 1x>1\dfrac1x>1 by "multiplying by xx": 1>x1>x. Is x=−5x=-5 a solution?

Section 2 of 14

Core idea

Start hereA trap in plain sight

Solve 1x>1\frac1x>1. Multiply by xx: 1>x1>x. So x=−5x=-5 works? But 1−5\frac1{-5} is negative! Multiplying by something that might be negative silently flips the inequality. Further maths has a neat fix.

The big idea

Multiply by the square of the denominator (always positive), move everything to one side, factorise, and read signs from a sketch. Exclude values where the original expression is undefined.

1. The method

Worked examplex+1x−2<2\frac{x+1}{x-2}<2

Solve.

Goal
Solution set.
Strategy
× (x−2)2(x-2)^2; factor out (x−2)(x-2).
  1. (x+1)(x−2)<2(x−2)2(x+1)(x-2)<2(x-2)^2
    (x−2)2>0(x-2)^2>0 for x≠2x\ne2.
  2. 0<(x−2)[2(x−2)−(x+1)]0<(x-2)\left[2(x-2)-(x+1)\right]
    Collect; common factor.
  3. 0<(x−2)(x−5)0<(x-2)(x-5)
    Simplify.
  4. x<2 or x>5x<2\text{ or }x>5
    U-shaped quadratic is positive outside its roots.
Answerx<2x<2 or x>5x>5.
✓ Checkx=0x=0: −12<2-\frac12<2 ✓; x=3x=3: 4<24<2 ✗ ✓.
Worked exampleA cubic sign pattern

Solve 2xx−1>x\frac{2x}{x-1}>x.

Goal
Solution.
Strategy
× (x−1)2(x-1)^2; factor x(x−1)x(x-1).
  1. 2x(x−1)>x(x−1)22x(x-1)>x(x-1)^2
    Multiply.
  2. 0>x(x−1)[(x−1)−2]=x(x−1)(x−3)0>x(x-1)\left[(x-1)-2\right]=x(x-1)(x-3)
    Collect.
  3. x<0 or 1<x<3x<0\text{ or }1<x<3
    Positive cubic is negative left of 00 and between 11 and 33.
Answerx<0x<0 or 1<x<31<x<3.
✓ Checkx=2x=2: 4>24>2 ✓; x=0.5x=0.5: −2>0.5-2>0.5 ✗ ✓.
Common misconceptionIncluding the asymptote In "≤\le" answers, never include the value making the denominator zero.
TrickGraphical check Sketch y=x+1x−2y=\frac{x+1}{x-2} and y=2y=2; the solution is where the curve is below the line.

Summary

  • Multiply by squares.
  • Factorise, sketch.
  • Exclude asymptotes.
Key words
Critical value
Where the expression is zero or undefined.
HistoryThe ">" sign. Thomas Harriot introduced the symbols >> and << in a book published in 1631, ten years after his death.
Fun factSign tables. A sign table is a tiny computer: track the sign of each factor across the number line and multiply. It never needs a graph.
Section 3 of 14

Key formulae

Key movePQ<k ⟹ PQ<kQ2(Q2>0)\frac{P}{Q}<k\ \Longrightarrow\ PQ<kQ^2\quad(Q^2>0)
Critical valueszeros of the factorised expression, including where Q=0\text{zeros of the factorised expression, including where }Q=0
TrickFactor, don't expand. After multiplying by Q2Q^2, take out the common factor QQ straight away: kQ2−PQ=Q(kQ−P)kQ^2-PQ=Q(kQ-P). Expanding everything wastes time.
Section 4 of 14

Worked examples

Example 10 / 3 steps
Solve x+1x−2<2\dfrac{x+1}{x-2}<2.
    Section 5 of 14

    Explore the graph

    Reading the solution from a graph
    Loading the interactive graph…
    Try this
    1. Where is the curve below the line y=2y=2? Check that it's x<2x<2 and x>5x>5.

    The curve meets y=2y=2 at x=5x=5 and jumps across at the asymptote x=2x=2.

    In the real worldSafe dosages. Drug concentration models are often rational functions of time; "when is the concentration above the effective level?" is a rational inequality.
    Section 6 of 14

    Common mistakes

    IncorrectMultiplying by x−2x-2
    CorrectMultiply by (x−2)2(x-2)^2
    x−2x-2 might be negative.
    IncorrectIncluding x=2x=2 in a ≤\le answer
    CorrectExclude it
    The expression is undefined there.
    IncorrectDividing both sides by a common factor
    CorrectKeep it as a factor
    You'd lose a critical value.

    Me: multiply both sides by (x − 2)

    The inequality when x < 2: flip! Use (x − 2)² instead.

    Section 7 of 14

    Exam tip

    Exam tipSketch the cubic or quadratic. After factorising, a quick sketch of the product (e.g. (x−2)(x−5)(x-2)(x-5)) shows immediately where it is positive or negative.
    Section 8 of 14

    Guided practice

    SA-FP-INEQ-G012 marksL2 StandardNon-calculatorRational inequalities
    Find the critical values when solving x+1x−2<2\dfrac{x+1}{x-2}<2.

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    Salam Academy Original Practice · Proposed mark scheme
    Section 9 of 14

    Practice questions

    SA-FP-INEQ-0013 marksL2 StandardNon-calculatorRational inequalities
    1x−1>1\dfrac1{x-1}>1 has solution a<x<ba<x<b. Find bb.

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    Salam Academy Original Practice · Proposed mark scheme
    SA-FP-INEQ-0023 marksL3 ExamNon-calculatorRational inequalities
    Find the smallest value of xx satisfying xx+1≥2\dfrac{x}{x+1}\ge2.

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    Salam Academy Original Practice · Proposed mark scheme
    SA-FP-INEQ-0032 marksL2 StandardNon-calculatorRational inequalities
    Find the smallest positive integer satisfying 3x+2≤1\dfrac3{x+2}\le1.

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    Salam Academy Original Practice · Proposed mark scheme
    SA-FP-INEQ-0042 marksL2 StandardNon-calculatorRational inequalities
    How many integers xx with −5≤x≤5-5\le x\le5 satisfy x−3x+1>0\dfrac{x-3}{x+1}>0?

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    Salam Academy Original Practice · Proposed mark scheme

    More questions on every skill in this subtopic are in the question bank under Practise.

    Section 10 of 14

    Challenge

    SA-FP-INEQ-0054 marksL4 ChallengingNon-calculatorRational inequalities
    Find the critical values for 2xx−1>x\dfrac{2x}{x-1}>x.

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    Salam Academy Original Practice · Proposed mark scheme
    Section 11 of 14

    Summary

    • Multiply by the square of the denominator.
    • Factorise; find critical values.
    • Sketch; exclude asymptotes.
    Section 12 of 14

    Quick quiz

    Q1 · SA-FP-INEQ-Q011 markL1 FoundationNon-calculatorRational inequalities
    Why multiply both sides by (x−a)2(x-a)^2 rather than (x−a)(x-a)?
    Choose one option

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    Salam Academy Original Practice · Proposed mark scheme
    Section 13 of 14

    Same topic, different board

    Specification
    7367 A-level Further Mathematics
    Where
    Further algebra and functions
    Expected
    Solve inequalities involving rational functions.
    Calculator
    Permitted on all papers.
    Typical style
    AQA mixes short multiple-choice questions with longer structured ones.
    Open in
    7367 · Paper 17367 · Paper 2
    Section 14 of 14

    Past paper questions

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    Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7367, First teaching 2017 (official specification).