Algebra: Manipulation, Equations and Inequalities
What you must command
Algebra questions test whether you can manipulate expressions cleanly and justify each step. Be secure with expanding brackets, including double brackets; factorising by common factors, difference of two squares and simple quadratics; simplifying algebraic fractions; solving linear equations and simultaneous equations by elimination and substitution; solving quadratics by factorising or the formula; rearranging formulae to change the subject; laws of indices; and solving linear inequalities, remembering to reverse the sign when multiplying or dividing by a negative number. Function notation and simple substitution also appear.
Traps that catch candidates
- A minus sign before a bracket changes every sign inside it: -(2x - 5) becomes -2x + 5.
- Squaring a bracket is not squaring each term: (x + 3)^2 is x^2 + 6x + 9.
- When solving, whatever you do to one side must be done to the other, and the same applies to every term.
- For inequalities, multiplying or dividing by a negative reverses the inequality sign.
- When changing the subject, undo operations in reverse order and keep the target variable alone at each stage.
- Always substitute your answer back into the original equation; it catches sign errors quickly.
Worked example
Solve 3(x - 2) = 5x + 4. Expand: 3x - 6 = 5x + 4. Subtract 3x: -6 = 2x + 4. Subtract 4: -10 = 2x, so x = -5. Check: left side 3(-7) = -21; right side 5(-5) + 4 = -21. Correct.
Rule of thumb
Treat an equation like a balance: record each operation on both sides on a fresh line. Neat lines earn method marks even when the final answer slips.