Solving Fractional-Rational Equations
📌 Definition
A fractional-rational equation is an equation that contains the variable in the denominator of one or more fractions. These are equations of the form:
$$\frac{P(x)}{Q(x)} = 0 \quad \text{or} \quad \frac{A(x)}{B(x)} = \frac{C(x)}{D(x)}$$
where $P(x), Q(x), A(x), B(x), C(x), D(x)$ are polynomials and the denominators are not zero.
⚠️ Key Concept: Domain Restrictions
Before solving, you must find the values of $x$ that make any denominator equal to zero. These values are called excluded values (or restrictions on the domain).
If a denominator is $Q(x)$, then we require:
$$Q(x) \neq 0$$
Any solution that equals an excluded value must be rejected — it is called an extraneous solution.
✅ Algorithm for Solving
- Find the domain restrictions — set each denominator ≠ 0
- Find the LCD (Least Common Denominator) of all fractions
- Multiply both sides of the equation by the LCD
- Simplify — you'll get a polynomial equation
- Solve the resulting equation
- Check each solution against domain restrictions