Overview
This lesson covers methods for solving rational equations, focusing on clearing fractions, finding common denominators, and using cross-multiplication and factoring techniques.
Clearing Fractions in Rational Equations
- To solve equations with fractions, multiply both sides by the least common multiple (LCM) of the denominators to eliminate fractions.
- Example: For ( \frac{5}{8} - \frac{3}{5} = \frac{x}{10} ), the LCM is 40.
Isolating the Variable
- Once fractions are cleared, solve for the variable using standard algebraic methods.
- Move all terms involving the variable to one side and constants to the other.
Cross-Multiplication Method
- For equations set up as two fractions equaling each other, use cross-multiplication to form a linear equation.
- Example: ( \frac{x+3}{x-3} = \frac{12}{3} ) leads to ( 12(x-3) = 3(x+3) ).
Solving Quadratic Equations
- Some rational equations, after clearing fractions, become quadratic equations.
- Factor the quadratic or use the square root property to find solutions.
- Always check for extraneous solutions.
Examples Covered
- Solving by multiplying both sides by x or LCM, then factoring.
- Cross-multiplication for two-fraction equations.
- Factoring difference of squares when denominators are polynomials (e.g., ( x^2-25 = (x+5)(x-5) )).
- Distributing, combining like terms, moving all terms to one side, then factoring.
Key Terms & Definitions
- Rational Equation — An equation containing one or more fractions with variables in the denominator.
- Least Common Multiple (LCM) — The smallest number that all denominators divide into evenly.
- Cross-Multiplication — Multiplying diagonally across a proportion to eliminate denominators.
- Extraneous Solution — A solution that does not satisfy the original equation after checking.
Action Items / Next Steps
- Practice more rational equation problems using LCM and cross-multiplication.
- Review factoring quadratics and difference of squares.
- Check all solutions in the original equations for extraneous answers.