Overview
This lecture explains how to solve rational equations by finding the least common denominator (LCD), clearing denominators, and solving for the variable, with several step-by-step examples.
Steps in Solving Rational Equations
- Start by identifying all the denominators in the rational equation.
- Find the least common denominator (LCD) that will clear all fractions.
- Multiply both sides of the equation by the LCD to eliminate denominators.
- Simplify the resulting equation to a linear or quadratic equation in standard form.
- Solve for the variable using algebraic methods.
- Check each solution by substituting back into the original equation to avoid extraneous solutions.
Finding the LCD
- For equations like 1/(2x) + 3/(5x) = 2/(x), the LCD is 10x.
- For denominators with variables, factor each denominator fully before finding the LCD.
- Examples: LCD of (x-1) and (x+2) is (x-1)(x+2); LCD of (x-1), (x+2), and (x-2) is (x-1)(x+2)(x-2).
Example Problems (Key Moments)
- Combine and solve rational equations by applying the LCD.
- Example: For 1/(x-1) + 2/(x+2) = 3/[(x-1)(x+2)], LCD is (x-1)(x+2).
- Multiply each term by the LCD, simplify and solve for x.
- Always check if the solutions are valid within the original equation's domain.
Application Example
- Rational equations can be applied to percentage problems, such as finding a number of games needed for a certain win percentage.
- Translate a statement into a rational equation, solve for the unknown, interpret the solution.
Key Terms & Definitions
- Rational Equation — An equation involving fractions with polynomials in the numerator and/or denominator.
- Least Common Denominator (LCD) — The smallest expression that is a common multiple of all denominators in the equation.
- Extraneous Solution — An apparent solution that does not satisfy the original equation due to restrictions from the denominators.
Action Items / Next Steps
- Practice solving rational equations by finding the LCD and clearing denominators.
- Attempt textbook exercises covering rational equations and their applications.
- Review steps for verifying solutions and identifying extraneous roots.