This module covers five ways an equation or inequality can need more care than a straight rearrangement: quadratic inequalities, equations with an absolute value, equations with square roots, exponential equations and equations with algebraic fractions. One idea connects them: a solving step can allow in answers the original question never accepted, so every solution needs a final check.
Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content, notation and calculator rules in your exam year. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to factorise quadratics, solve linear and quadratic equations, and use index laws with whole-number powers. If your quadratics feel shaky, revise quadratic structure and discriminants first, because the roots and their signs carry the whole inequality lesson.
An orienting example
Solve |x − 1| = 2x − 5.
Step 1, two cases: x − 1 = 2x − 5, or x − 1 = −(2x − 5).
Step 2, case 1: x − 1 = 2x − 5 gives x = 4.
Step 3, case 2: x − 1 = −2x + 5 gives 3x = 6, so x = 2.
Step 4, check x = 4: |3| = 3 and 2(4) − 5 = 3. It works.
Step 5, check x = 2: |1| = 1 but 2(2) − 5 = −1. A modulus cannot equal a negative number, so x = 2 fails.
Answer: x = 4 only.
Notice what happened. Correct algebra produced two candidates, and only the check separated the real answer from the false one. The same pattern repeats through the whole module.
In which order should you study it?
- Solve a quadratic inequality using sign intervals: the roots-and-regions idea that also underpins later sign work.
- Solve an equation involving an absolute value: splitting into cases, with the first real test of checking answers.
- State restrictions before manipulating fractions: recording what x may not be before you start.
- Reject an extraneous root after squaring: the clearest case of a false root appearing from a valid step.
- Solve an exponential equation by a common base: index laws turned into linear equations, with a check at the end.
What are the common traps?
- Writing a two-piece inequality answer as a single chain, such as 3 ≤ x ≤ −1/2.
- Negating only part of the right side in a modulus case.
- Stopping at the quadratic after squaring and reporting both roots.
- Comparing exponents while the bases are still different.
- Multiplying through by a denominator before writing the restriction.
Each trap is avoided by the same habit: finish by substituting back or comparing with your restriction list.
How should you use the practice set?
Work through the mixed practice set after the lessons, on paper, with full working. Questions are ordered from easier to harder and they are mixed on purpose, so you practise choosing the method, not only applying it. The quadratic structure explorer helps you see inequality regions, and the non-calculator working trainer supports the exact checks.
Students who like to work through the topic with direct feedback on their own working can ask about online one-to-one Additional Mathematics tuition.