Worked solutions: Complex rational equations with identified extraneous values

📚 Understanding Complex Rational Equations

Complex rational equations involve fractions where the numerator and/or denominator contain variables. Solving these equations requires careful manipulation to eliminate the fractions and isolate the variable. A critical aspect is identifying extraneous solutions, which are potential solutions that do not satisfy the original equation.

📜 Historical Context

The study of rational equations dates back to ancient civilizations, where mathematicians grappled with problems involving proportions and ratios. The development of algebraic notation in the 16th and 17th centuries provided a more systematic way to solve these equations. The concept of extraneous solutions became increasingly important as algebraic techniques advanced, highlighting the need for verification in equation solving.

🔑 Key Principles for Solving Rational Equations

• 🔍 Finding the Least Common Denominator (LCD): This is the smallest multiple that all denominators divide into. It's crucial for clearing fractions.
• 💡 Multiplying by the LCD: Multiplying both sides of the equation by the LCD eliminates the fractions, simplifying the equation.
• 📝 Solving the Resulting Equation: After clearing fractions, solve the resulting polynomial equation using standard algebraic techniques.
• ✔️ Checking for Extraneous Solutions: Always substitute the solutions back into the original equation to ensure they are valid. Extraneous solutions occur when a solution makes a denominator equal to zero.

⚙️ Step-by-Step Solution Process

1. Factor all denominators: Factoring helps in identifying the LCD.
2. Identify the LCD: Determine the least common denominator of all fractions in the equation.
3. Multiply both sides by the LCD: This eliminates the fractions.
4. Solve the resulting equation: Simplify and solve for the variable.
5. Check for extraneous solutions: Substitute each solution back into the original equation. Discard any solution that makes a denominator zero.

➗ Example 1: Solving a Rational Equation

Solve for $x$ in the equation: $\frac{2}{x-3} + \frac{1}{x} = \frac{5}{4x}$

1. Identify the LCD: The LCD is $4x(x-3)$.
2. Multiply by the LCD: Multiply both sides of the equation by $4x(x-3)$: $4x(x-3) \cdot \frac{2}{x-3} + 4x(x-3) \cdot \frac{1}{x} = 4x(x-3) \cdot \frac{5}{4x}$
3. Simplify: $8x + 4(x-3) = 5(x-3)$
4. Expand and solve: $8x + 4x - 12 = 5x - 15$ $12x - 12 = 5x - 15$ $7x = -3$ $x = -\frac{3}{7}$
5. Check for extraneous solutions: Substitute $x = -\frac{3}{7}$ back into the original equation. Since none of the denominators are zero, $x = -\frac{3}{7}$ is a valid solution.

🤯 Example 2: Identifying Extraneous Solutions

Solve for $x$ in the equation: $\frac{x}{x-2} = \frac{2}{x-2} + 2$

1. Identify the LCD: The LCD is $(x-2)$.
2. Multiply by the LCD: Multiply both sides of the equation by $(x-2)$: $(x-2) \cdot \frac{x}{x-2} = (x-2) \cdot \frac{2}{x-2} + 2(x-2)$
3. Simplify: $x = 2 + 2x - 4$
4. Solve: $x = 2 + 2x - 4$ $x = -2 + 2x$ $-x = -2$ $x = 2$
5. Check for extraneous solutions: Substitute $x = 2$ back into the original equation. The denominators become zero, so $x = 2$ is an extraneous solution. Therefore, there is no solution to this equation.

✍️ Practice Quiz

Solve the following rational equations and identify any extraneous solutions:

1. $\frac{3}{x} + \frac{1}{2} = \frac{4}{x}$
2. $\frac{x}{x+1} = \frac{1}{x+1} + 1$
3. $\frac{2}{x-1} = \frac{4}{x+1}$
4. $\frac{1}{x} + \frac{1}{x+1} = \frac{5}{6}$
5. $\frac{x+3}{x-2} = \frac{5}{x-2}$
6. $\frac{x}{x-5} = \frac{5}{x-5} + 3$
7. $\frac{1}{x-2} + \frac{1}{x+2} = \frac{4}{x^2-4}$

💡 Tips and Tricks

• 🧪 Always Factor: Factoring denominators makes it easier to find the LCD.
• 🔢 Double-Check: Verify each solution in the original equation.
• 🌍 LCD First: Multiplying by the LCD early simplifies the process.

📈 Advanced Techniques

For more complex equations, consider using substitution to simplify the equation. Also, be aware of quadratic equations that may arise after clearing fractions, and use factoring or the quadratic formula to solve them.

🔑 Conclusion

Solving complex rational equations requires a systematic approach, including finding the LCD, clearing fractions, solving the resulting equation, and, crucially, checking for extraneous solutions. By following these steps and understanding the underlying principles, you can confidently tackle these types of problems.