🔄 Rational Expressions & Equations — Domains & Extraneous Roots
Essential for fraction problems and equation solving in AMC contests.
🎯 Key Ideas
Rational Expressions — Fractions with polynomial numerators and denominators. Domain restrictions occur when denominators equal zero.
Rational Equations — Equations involving rational expressions, solved by cross-multiplying and checking for extraneous solutions.
Domain Restrictions — Values that make denominators zero must be excluded from the domain.
📊 Essential Concepts
Domain Restrictions
| Expression | Domain Restriction | Example |
|---|---|---|
| $\frac{1}{x}$ | $x \neq 0$ | $x \in \mathbb{R} \setminus {0}$ |
| $\frac{1}{x-2}$ | $x \neq 2$ | $x \in \mathbb{R} \setminus {2}$ |
| $\frac{x+1}{x^2-4}$ | $x \neq \pm 2$ | $x \in \mathbb{R} \setminus {-2, 2}$ |
Solving Rational Equations
| Step | Action | Example |
|---|---|---|
| 1 | Find common denominator | $\frac{1}{x} + \frac{1}{x-1} = \frac{2}{x(x-1)}$ |
| 2 | Cross-multiply | $x(x-1) \cdot \frac{1}{x} + x(x-1) \cdot \frac{1}{x-1} = x(x-1) \cdot \frac{2}{x(x-1)}$ |
| 3 | Simplify | $(x-1) + x = 2$ |
| 4 | Solve | $2x - 1 = 2$ → $x = \frac{3}{2}$ |
| 5 | Check domain | $x = \frac{3}{2} \neq 0, 1$ ✓ |
🎯 Micro-Examples
Example 1: Simplify $\frac{x^2 - 4}{x^2 - 2x}$
- Factor numerator: $x^2 - 4 = (x-2)(x+2)$
- Factor denominator: $x^2 - 2x = x(x-2)$
- Simplify: $\frac{(x-2)(x+2)}{x(x-2)} = \frac{x+2}{x}$ (for $x \neq 2$)
- Domain: $x \neq 0, 2$
Example 2: Solve $\frac{1}{x} + \frac{1}{x-1} = \frac{2}{x(x-1)}$
- Domain: $x \neq 0, 1$
- Cross-multiply: $(x-1) + x = 2$
- Simplify: $2x - 1 = 2$
- Solve: $x = \frac{3}{2}$
- Check: $\frac{3}{2} \neq 0, 1$ ✓
Example 3: Solve $\frac{x+1}{x-2} = \frac{x-3}{x+4}$
- Domain: $x \neq 2, -4$
- Cross-multiply: $(x+1)(x+4) = (x-3)(x-2)$
- Expand: $x^2 + 5x + 4 = x^2 - 5x + 6$
- Simplify: $10x = 2$
- Solve: $x = \frac{1}{5}$
- Check: $\frac{1}{5} \neq 2, -4$ ✓
⚠️ Common Traps & Fixes
Trap: Forgetting to check domain restrictions
- Fix: Always identify values that make denominators zero
- Example: $\frac{1}{x-2}$ requires $x \neq 2$
Trap: Not checking for extraneous solutions
- Fix: Always substitute solutions back into original equation
- Example: Cross-multiplying can introduce extraneous solutions
Trap: Simplifying incorrectly
- Fix: Factor completely before canceling
- Example: $\frac{x^2-4}{x^2-2x} = \frac{(x-2)(x+2)}{x(x-2)} = \frac{x+2}{x}$ (not $\frac{x+2}{x-2}$)
🎯 AMC-Style Worked Example
Problem: Find all real solutions to $\frac{x^2 + x - 6}{x^2 - 4} = \frac{x - 2}{x + 2}$.
Solution:
- Factor numerator: $x^2 + x - 6 = (x+3)(x-2)$
- Factor denominator: $x^2 - 4 = (x-2)(x+2)$
- Rewrite equation: $\frac{(x+3)(x-2)}{(x-2)(x+2)} = \frac{x-2}{x+2}$
- Simplify left side: $\frac{x+3}{x+2} = \frac{x-2}{x+2}$ (for $x \neq 2$)
- Cross-multiply: $(x+3)(x+2) = (x-2)(x+2)$
- Expand: $x^2 + 5x + 6 = x^2 - 4$
- Simplify: $5x + 6 = -4$
- Solve: $5x = -10$, so $x = -2$
- Check domain: $x = -2$ makes denominator zero
- Answer: No real solutions (extraneous solution)
Key insight: Always check domain restrictions and verify solutions.
🔗 Related Topics
- Factoring — Essential for simplifying rational expressions
- Domain — Rational expressions have domain restrictions
- Extraneous Solutions — Cross-multiplying can introduce extra solutions
- Word Problems — Rational equations often model real-world situations
📝 Practice Checklist
- Master domain restrictions
- Practice simplifying rational expressions
- Learn to solve rational equations
- Practice checking for extraneous solutions
- Understand cross-multiplying technique
- Work on speed and accuracy
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