O-Level E-Math and SEC G3 Mathematics K310
N7: Equations and inequalities
Choose an algebraic or graphical route, preserve inequality direction, and check solutions in the original condition.
Core notes
Solving an equation means finding values that make the original statement true. Choose a method suited to the equation, preserve restrictions, and substitute back when an operation may have introduced or hidden a solution.
Linear and fractional equations
Use inverse operations while keeping both sides balanced. For a fractional equation, state values that make a denominator zero, multiply by the lowest common denominator, solve the resulting equation and reject excluded values.
Simultaneous equations
Substitution is efficient when one variable is already isolated. Elimination is efficient when coefficients can be matched. Graphically, the solution is the intersection of the two lines, so a read-off answer may be approximate.
Quadratic equations
Factorisation exposes roots when suitable factors exist. The quadratic formula works generally when . Completing the square reveals the turning-point structure, while a graph gives the -coordinates of intersections with the horizontal axis.
Modelling and inequalities
Define the unknown, translate the conditions into an equation or inequality, solve, then interpret the answer in context. Represent a linear inequality on a number line with the correct open or closed endpoint.
Formulae and methods
| Relationship | Use |
|---|---|
| Solve . |
Worked examples
Example 1: Solve .
- Factorise as .
- Set each factor equal to zero.
- Obtain or .
Answer: or .
Example 2: Solve .
- Subtract : .
- Divide by and reverse the inequality.
- Write .
Answer: .
Chapter checkpoint
- State denominator restrictions before clearing fractions.
- Choose factorisation, formula, completion of the square or a graph according to the quadratic form.
- For inequalities, reverse the sign after multiplying or dividing by a negative number.
Exam traps and retrieval check
Avoid these traps
- Clearing denominators without recording excluded values.
- Forgetting both roots of a quadratic equation.
- Failing to reverse an inequality after division by a negative number.
Check from memory
What does a graphical simultaneous solution represent?
The intersection point of the two graphs.
When is the quadratic formula valid?
For with .
What endpoint represents on a number line?
An open circle at .
Official K310 coverage
This chapter maps to 7 official outcomes: N7.1, N7.2, N7.3, N7.4, N7.5, N7.6, N7.7. Check the official 2027 K310 syllabus for exact assessable wording and paper details.