Page 43 - HSC Inequalities

1. HSC-Inequalities - an Introduction

Inequalities represent a significant paradigm shift from standard equations; instead of searching for a single right answer, students must navigate a continuum of valid solutions. In the HSC (particularly Extension 2), inequalities are notoriously tricky because they test a student's ability to reason graphically and algebraically simultaneously. While a linear inequality is simple, absolute value inequalities, rational inequalities (fractions), and the famous AM-GM (Arithmetic Mean-Geometric Mean) inequality require immense care. A single division by a negative number—or an unknown variable—can flip the sign and derail an entire proof.

2. Learning Outcomes & Syllabus Mapping

• Solve absolute value inequalities and graph regions.
• Solve rational inequalities by multiplying by the square of the denominator.
• The Nature of Proof (using AM-GM, triangle inequality, and deductive reasoning to prove abstract inequalities).

3. Prerequisites

• Solid understanding of graphing and transformations (Functions).
• Familiarity with factorization and completing the square.
• Understanding of basic logic and proof techniques.

4. Common HSC Mistakes

The cardinal sin of HSC inequalities is multiplying or dividing both sides of an inequality by a variable expression (like $x$ or $x-2$) without knowing its sign, completely ignoring the fact that a negative value flips the inequality symbol. Students also frequently struggle to find the correct intersection or union of domains when dealing with double inequalities or absolute values.

5. Sample Worked Problem

6. Exam Strategy & Weighting

Solving a standard rational or absolute value inequality is a common feature early in the Extension 1 paper. In Extension 2, inequality proofs form a major component of the "Nature of Proof" topic, often appearing in the more challenging Questions 15 or 16. The geometric approach—sketching both sides of the inequality to see which graph lies "above" the other—is a highly recommended time-saving strategy for multiple-choice questions.

7. Key Definitions / Glossary Summary

• Rational Inequality: An inequality containing a polynomial fraction (e.g., $\frac{P(x)}{Q(x)} > 0$).
• Critical Points: The $x$-values where an expression equals zero or is undefined (vertical asymptotes).
• AM-GM Inequality: For non-negative numbers, the Arithmetic Mean is always greater than or equal to the Geometric Mean (e.g., $\frac{a+b}{2} \ge \sqrt{ab}$).

8. Frequently Asked Questions (FAQ)

Q: Can I solve rational inequalities using a sign diagram instead of multiplying by the square? A: Yes, finding the critical points and testing values in regions on a number line (the test-point method) is perfectly valid and often faster for complex fractions.

Q: Will I need to prove the AM-GM inequality? A: Yes, proving the base case for two variables ($a$ and $b$) by expanding $(\sqrt{a} - \sqrt{b})^2 \ge 0$ is a standard Extension 2 requirement.

9. Where to next?

Mastering inequalities gives you the exact graphical and algebraic toolkit needed to succeed in advanced curve sketching. Move on to the HSC-Functions booklet to refine your graphing, or dive into HSC-Proofs to tackle formal Extension 2 inequality logic.