⚖️ Inequalities & Optimization — Sign Charts & AM-GM
Essential for optimization problems and advanced inequality solving in AMC contests.
🎯 Key Ideas
Sign Charts — Visual method for solving polynomial inequalities by analyzing sign changes at roots.
AM-GM Inequality — For positive real numbers, $\frac{a+b}{2} \geq \sqrt{ab}$ with equality when $a = b$.
Optimization — Finding maximum or minimum values using algebraic techniques and inequalities.
📊 Essential Concepts
Sign Charts
| Step | Action | Example |
|---|---|---|
| 1 | Find roots | $x^2 - 4 = 0$ → $x = \pm 2$ |
| 2 | Create number line | $-\infty$ ——— $-2$ ——— $2$ ——— $\infty$ |
| 3 | Test intervals | $x < -2$: $(-)^2 - 4 > 0$; $-2 < x < 2$: $(-)^2 - 4 < 0$; $x > 2$: $(+)^2 - 4 > 0$ |
| 4 | Write solution | $x^2 - 4 > 0$ when $x < -2$ or $x > 2$ |
AM-GM Inequality
| Form | Statement | Example |
|---|---|---|
| Basic | $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b > 0$ | $\frac{4+9}{2} = 6.5 \geq \sqrt{36} = 6$ |
| General | $\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}$ | $\frac{2+3+6}{3} = \frac{11}{3} \geq \sqrt[3]{36} \approx 3.3$ |
| Equality | Holds when all numbers are equal | $a = b = c$ in 3-variable case |
🎯 Micro-Examples
Example 1: Solve $x^2 - 4 > 0$
- Find roots: $x^2 - 4 = 0$ → $x = \pm 2$
- Create sign chart: $-\infty$ ——— $-2$ ——— $2$ ——— $\infty$
- Test intervals:
- $x < -2$: $(-3)^2 - 4 = 9 - 4 = 5 > 0$ ✓
- $-2 < x < 2$: $0^2 - 4 = -4 < 0$ ✗
- $x > 2$: $3^2 - 4 = 9 - 4 = 5 > 0$ ✓
- Answer: $x < -2$ or $x > 2$
Example 2: Find minimum value of $x + \frac{1}{x}$ for $x > 0$
- Apply AM-GM: $\frac{x + \frac{1}{x}}{2} \geq \sqrt{x \cdot \frac{1}{x}} = 1$
- Solve: $x + \frac{1}{x} \geq 2$
- Equality when: $x = \frac{1}{x}$ → $x^2 = 1$ → $x = 1$ (since $x > 0$)
- Answer: Minimum value is $2$ when $x = 1$
Example 3: Solve $(x-1)(x+2)(x-3) \leq 0$
- Find roots: $x = 1, -2, 3$
- Create sign chart: $-\infty$ ——— $-2$ ——— $1$ ——— $3$ ——— $\infty$
- Test intervals:
- $x < -2$: $(-)(-)(-) = -$ ✗
- $-2 < x < 1$: $(-)(+)(-) = +$ ✓
- $1 < x < 3$: $(+)(+)(-) = -$ ✗
- $x > 3$: $(+)(+)(+) = +$ ✓
- Answer: $-2 \leq x \leq 1$ or $x \geq 3$
⚠️ Common Traps & Fixes
Trap: Forgetting to check equality at roots
- Fix: Include roots in solution if inequality allows equality
- Example: $x^2 - 4 \geq 0$ includes $x = \pm 2$
Trap: Incorrect sign analysis
- Fix: Test one value in each interval, not just the sign of leading coefficient
- Example: For $(x-1)(x+2)$, test $x = 0$: $(0-1)(0+2) = (-1)(2) = -2 < 0$
Trap: AM-GM without positive constraint
- Fix: AM-GM only applies to positive numbers
- Example: Can’t use AM-GM on $x + \frac{1}{x}$ if $x < 0$
🎯 AMC-Style Worked Example
Problem: Find the minimum value of $x^2 + \frac{1}{x^2}$ for $x > 0$.
Solution:
- Apply AM-GM: $\frac{x^2 + \frac{1}{x^2}}{2} \geq \sqrt{x^2 \cdot \frac{1}{x^2}} = 1$
- Solve: $x^2 + \frac{1}{x^2} \geq 2$
- Equality when: $x^2 = \frac{1}{x^2}$ → $x^4 = 1$ → $x = 1$ (since $x > 0$)
- Verify: When $x = 1$: $1^2 + \frac{1}{1^2} = 1 + 1 = 2$
- Answer: Minimum value is $2$ when $x = 1$
Key insight: AM-GM is powerful for finding lower bounds on positive expressions.
🔗 Related Topics
- Quadratic Inequalities — Sign charts work for any polynomial
- Optimization — AM-GM is essential for many optimization problems
- Domain — Inequalities often involve domain restrictions
- Word Problems — Optimization problems often use inequalities
📝 Practice Checklist
- Master sign chart technique
- Practice AM-GM inequality
- Learn optimization strategies
- Practice polynomial inequalities
- Understand equality conditions
- Work on speed and accuracy
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