Inequalities — A/L Combined Mathematics

What must I remember about Inequalities?

• Properties of inequalities: If a > b and c > 0, then ac > bc; if a > b and c < 0, the inequality sign REVERSES: ac < bc; if a > b, then a + c > b + c.
• Sign chart / Wavy curve method for rational inequalities: Express in form P(x)/Q(x) > 0 or < 0; find critical points (roots of P(x) = 0 and Q(x) = 0), mark on real number line, and determine alternating signs across intervals.
• Modulus / Absolute value inequalities: |x| < a <=> -a < x < a (for a > 0); |x| > a <=> x > a or x < -a; |f(x)| < g(x) <=> -g(x) < f(x) < g(x) with g(x) > 0.
• Squaring both sides rule: a > b => a² > b² is VALID if and only if both a ≥ 0 and b ≥ 0; squaring negative quantities or terms of unknown sign without casework leads to false solutions.
• Never cross-multiply by variables: In P(x)/Q(x) > c, NEVER multiply by Q(x) directly unless Q(x) is known to be strictly positive (e.g., Q(x) = x² + 1); instead, subtract c and express over common denominator or multiply by [Q(x)]².

Test yourself (1 minute)

1. Strict exclusion of denominator zeros

   The solutions of P(x)/Q(x) ≥ 0 can include roots where P(x) = 0, but MUST STRICTLY EXCLUDE points where Q(x) = 0 to prevent division by zero.

2. Triangle Inequality

   For all real numbers x and y, |x + y| ≤ |x| + |y|; equality holds if and only if xy ≥ 0 (x and y have the same sign or at least one is zero).

3. AM-GM Inequality

   For non-negative real numbers a and b, (a + b)/2 ≥ √(ab); equality holds if and only if a = b; useful for finding minimum/maximum values.

Practise Inequalities questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 7

What should I study first?

Next in the syllabus

Where does Inequalities fit in the syllabus?

Year 1 · Term 1