Q: What does H2 Maths Notes (JC 1-2): 1.3) Equations and Inequalities cover? A: Algebraic, graphical, and numerical techniques for solving equations and inequalities in H2 Maths Topic 1.3.
Before you revise Practise each solving technique with and without technology. Mark schemes expect both algebraic reasoning and clear statements of graphing calculator (GC) commands used when numerical methods are required.
Equation-Solving Toolkit
Exact algebraic methods: factorisation, substitution, completing the square, and standard formulae.
Graphical methods: plot y=f(x) and y=g(x); solutions satisfy f(x)=g(x)
.
Iterative methods: rearrange into x=g(x) or use Newton-Raphson xn+1=xn−f′(xn)f(xn).
Numerical equation solver: graphing calculator (GC) features (SOLVE, ISCT, nSolve) find roots when analytic solutions are messy; document the initial guess or interval.
Example -- Iteration
Solve x=cosx.
Define g(x)=cosx; iterate xn+1=g(xn).
With x0=0.7, the sequence converges to 0.739085133….
State the termination criterion (difference less than 10−4).
Convergence note: here g′(x)=−sinx so ∣g′(x)∣=∣sinx∣<1 near the fixed point, which ensures local convergence of the iteration.
Polynomial and Rational Equations
For quadratics, use the discriminant b2−4ac to comment on nature of roots.
Cubics: attempt factor theorem or GC ISCT to guess rational roots before long division.
Rational equations require clearing denominators carefully and checking for extraneous roots.
Example -- Rational equation
Solve x−1x+2=3.
Multiply both sides by x−1: x+2=3x−3.
Simplify to 2x=5⇒x=25.
Ensure x=1 (domain restriction).
Inequalities
Linear and Polynomial Forms
Rearrange to f(x)>0 or f(x)≤0 and sign-test intervals around roots.
Quadratics: identify roots α,β and sketch or use sign chart.
Example -- Quadratic inequality
Solve x2−5x+6≥0.
Factor: (x−2)(x−3)≥0.
Sign chart shows positive on (−∞,2]∪[3,∞).
Rational Inequalities
Identify critical points from numerator and denominator.
