A rational equation is an equation that involves at least one rational expression, which is the ratio of two polynomials. Rational equations often look like fractions with variables in the denominator. Here’s an example of a rational equation:

$\frac{2x + 1}{x - 3} = \frac{4x}{x + 2}$

Steps to Solve a Rational Equation:

1. Identify the LCD (Lowest Common Denominator): Find the common denominator for all the rational expressions.
2. Multiply through by the LCD: This clears the denominators.
3. Simplify and solve the resulting equation.
4. Check for extraneous solutions: Any solution that makes the original denominator zero is not valid.

Example:

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

1. LCD: $(x+1)(x-1)$
2. Multiply both sides by the LCD: $2(x-1) + 3(x+1) = 5(x+1)(x-1)$
3. Expand and simplify: $2x - 2 + 3x + 3 = 5(x^2 - 1)$ $5x + 1 = 5x^2 - 5$
4. Move all terms to one side: $0 = 5x^2 - 5x - 6$
5. Solve the quadratic using factoring or the quadratic formula: $x = 2, x = -\frac{3}{5}$
6. Check for extraneous solutions by plugging the solutions back into the original equation.

Would you like to go through a similar example step-by-step, or upload a specific equation you’re working on?

Related Questions:

1. What is a rational function, and how does it differ from a rational equation?
2. How do you determine the domain of a rational equation?
3. Can rational equations have no solutions?
4. How do you solve rational equations with complex denominators?
5. What are common pitfalls when solving rational equations?

Tip: Always check for solutions that make the denominator zero, as these are invalid!