Inequalities

    OCR
    GCSE

    This topic necessitates the rigorous solution of inequalities involving rational functions and modulus arguments, extending significantly beyond linear and quadratic forms. Candidates must demonstrate mastery of algebraic techniques, specifically multiplying by the square of the denominator to preserve inequality direction, or squaring both sides to resolve modulus terms. The integration of graphical analysis is mandatory to identify critical values, vertical asymptotes, and intersection points, ensuring the accurate definition of solution regions using set or interval notation.

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    Objectives
    3
    Exam Tips
    4
    Pitfalls
    4
    Key Terms
    5
    Mark Points

    What You Need to Demonstrate

    Key skills and knowledge for this topic

    • Award M1 for a correct method to solve the associated quadratic equation (factorising, formula, or completing the square) to find critical values
    • Award A1 for identifying both correct critical values (e.g., x = -3, x = 5)
    • Award M1 for a method to determine the correct region, such as a sketch of the parabola or testing values between/outside roots
    • Award A1 for the correct final inequality notation; accept set notation {x : x < -3} ∪ {x : x > 5} or simple inequalities
    • Deduct 1 mark for 'run-on' inequalities in disjoint regions (e.g., 5 < x < -3)

    Example Examiner Feedback

    Real feedback patterns examiners use when marking

    • "You have correctly calculated the critical values, but your final answer describes the wrong region. Draw a sketch to visualize where the graph is above/below the axis."
    • "Avoid writing '5 < x < 2' for outside regions. This implies x is greater than 5 AND less than 2, which is impossible. Use 'x < 2 OR x > 5'."
    • "Check the question for 'distinct' vs 'real' roots—this determines whether you use > or ≥ in your discriminant inequality."
    • "Excellent use of the quadratic formula. To secure the final mark, ensure your inequality signs match the strictness of the original question."

    Marking Points

    Key points examiners look for in your answers

    • Award M1 for a correct method to solve the associated quadratic equation (factorising, formula, or completing the square) to find critical values
    • Award A1 for identifying both correct critical values (e.g., x = -3, x = 5)
    • Award M1 for a method to determine the correct region, such as a sketch of the parabola or testing values between/outside roots
    • Award A1 for the correct final inequality notation; accept set notation {x : x < -3} ∪ {x : x > 5} or simple inequalities
    • Deduct 1 mark for 'run-on' inequalities in disjoint regions (e.g., 5 < x < -3)

    Examiner Tips

    Expert advice for maximising your marks

    • 💡Always sketch the quadratic curve. Even a rough sketch showing the orientation (U-shape or n-shape) and roots ensures you select the correct 'inside' or 'outside' region.
    • 💡When solving discriminant questions (e.g., 'find k for distinct real roots'), remember the resulting expression is often a quadratic inequality in k.
    • 💡Never multiply or divide by a variable term (like x) unless you know its sign, as this flips the inequality symbol; bring all terms to one side instead.

    Common Mistakes

    Pitfalls to avoid in your exam answers

    • Writing mathematically impossible 'run-on' inequalities for disjoint regions, such as 4 < x < -2, instead of two separate statements
    • Stopping at the critical values (treating it as an equation) and failing to identify the inequality region
    • Incorrectly manipulating the inequality without first rearranging to ax² + bx + c > 0, leading to sign errors
    • Confusing strict (<) and non-strict (≤) inequalities, particularly when interpreting the discriminant condition for 'real roots' (b² - 4ac ≥ 0)

    Study Guide Available

    Comprehensive revision notes & examples

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    Key Terminology

    Essential terms to know

    Rational inequalities involving algebraic fractions
    Modulus inequalities and geometric interpretation
    Critical values and asymptotic behaviour
    Set notation and interval notation for solution regions

    Likely Command Words

    How questions on this topic are typically asked

    Solve
    Find the set of values
    Determine
    Sketch
    Show that

    Practical Links

    Related required practicals

    • •{"code":"Kinematics","title":"Time intervals in projectile motion","relevance":"Determining the time interval for which a projectile is above a certain height (h > 10)"}
    • •{"code":"Geometry","title":"Area constraints","relevance":"Finding the range of values for x for which a shape's area exceeds a specific value"}

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