H2 Maths Notes (JC 1-2): 1.3) Equations and Inequalities

Before you revise\ Practise each solving technique with and without technology. Mark schemes expect both algebraic reasoning and clear statements of 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 \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \).
• Numerical equation solver: GC features (SOLVE, ISCT, nSolve) find roots when analytic solutions are messy; document the initial guess or interval.

Example -- Iteration

Solve \( x = \cos x \).

1. Define \( g(x) = \cos x \); iterate \( x_{n+1} = g(x_n) \).
2. With \( x_0 = 0.7 \), the sequence converges to \( 0.739,085,133… \).
3. State the termination criterion (difference less than \( 10^-4 \)).

Polynomial and Rational Equations

• For quadratics, use the discriminant \( b^2 - 4ac \) 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 \( \frac{x + 2}{x - 1} = 3 ).

1. Multiply both sides by \( x - 1 \): \( x + 2 = 3x - 3 \).
2. Simplify to \( 2x = 5 \Rightarrow x = \tfrac{5}{2} \).
3. Ensure \( x \neq 1 \) (domain restriction).

Inequalities

Linear and Polynomial Forms

• Rearrange to \( f(x) > 0 \) or \( f(x) \leq 0 \) and sign-test intervals around roots.
• Quadratics: identify roots \( \alpha, \beta \) and sketch or use sign chart.

Example -- Quadratic inequality

Solve \( x^2 - 5x + 6 \geq 0 \).

1. Factor: \( (x - 2)(x - 3) \geq 0 \).
2. Sign chart shows positive on \( (-\infty, 2] \cup [3, \infty) \).

Rational Inequalities

• Identify critical points from numerator and denominator.
• Construct sign tables including excluded values.
• Express solution sets using interval notation.

Example -- Rational inequality

Solve \( \frac{2x - 5}{x + 3} < 1 \).

1. Rearrange: \( \frac{2x - 5}{x + 3} - 1 < 0 \) (simplify to) \( \frac{x - 8}{x + 3} < 0 \).
2. Critical points at \( x = -3 \) (excluded) and \( x = 8 \).
3. Sign chart yields solution \( (-3, 8) \), excluding \( x = -3 \).

Inequalities with Modulus and Exponentials

• Split modulus cases: \( \lvert f(x) \rvert < a \) implies \( -a < f(x) < a \).
• For exponentials/logarithms, apply monotonicity: \( a^x > a^y \iff x > y \) for \( a > 1 \).

Simultaneous Equations

• Linear systems: solve via elimination, substitution, or matrix methods. Express solutions clearly (point, line, or no solution).
• Linear-quadratic or quadratic-quadratic systems: substitute solutions from one equation into the other.

Example -- Line-circle intersection

Find intersections of \( y = 2x - 1 \) with \( x^2 + y^2 = 13 \).

1. Substitute: \( x^2 + (2x - 1)^2 = 13 \) giving \( 5x^2 - 4x - 12 = 0 \).
2. Solve: \( x = \frac{4 \pm \sqrt{16 + 240}}{10} = \frac{4 \pm \sqrt{256}}{10} = \frac{4 \pm 16}{10} \).
3. Hence \( x = 2 \) or \( x = -\tfrac{6}{5} \, y = 3 \) or \( y = -\tfrac{17}{5} \).

Numerical Methods and Technology

• Newton-Raphson: ensure derivative \( f'(x_n) \neq 0 \); document iteration formula and starting value.
• Fixed-point iteration: check convergence using \( \lvert g'(x) \rvert < 1 \) near the root.
• State GC commands (e.g. Solver, nSolve, G-Solv) and the viewing window or guesses used.

Example -- Newton-Raphson

Find a root of \( f(x) = x^3 - x - 1 \) to 3 significant figures.

1. Choose \( x_0 = 1 \).
2. Iteration: \( x_{n+1} = x_n - \frac{x_n^3 - x_n - 1}{3x_n^2 - 1} \).
3. Sequence: \( x_1 = 1.5 \, x_2 = 1.347,826 \, x_3 = 1.324,718 \).
4. Root \( \approx 1.32 \) to 3 s.f.

Exam Watch Points

• State the domain restrictions and justify exclusion of invalid roots.
• For inequalities, present the solution set clearly and include interval notation or number line sketches.
• When using numerical methods, write the iteration formula, initial guess, and termination condition.
• Comment on convergence (or failure) if a method does not work.

Quick Revision Checklist

• [ ] Switch between algebraic, graphical, and numerical methods appropriately.
• [ ] Construct accurate sign charts for polynomials and rational expressions.
• [ ] Explain the logic behind iteration methods, including convergence checks.
• [ ] Verify solutions against original equations to avoid extraneous roots.

Next steps: Move into Topic 2.1 to master sequences, recursions, and series manipulation.