Because division by zero is undefined, any value of the variable that makes the denominator equal to zero must be excluded from the domain.

Multiplication and Division:

• Multiplication: Factor all numerators and denominators completely, then cancel common factors before multiplying across.
• Division: Multiply the first expression by the reciprocal of the second expression (flip the second fraction), then proceed as multiplication.

Addition and Subtraction:

• Same Denominators: Add or subtract the numerators and keep the common denominator.
• Different Denominators: Find the Least Common Denominator (LCD), convert each fraction to an equivalent fraction with the LCD, then add/subtract numerators.

Simplification Strategies:

• Method 1 (Single Fraction): Simplify the numerator into a single fraction and the denominator into a single fraction. Then multiply the top fraction by the reciprocal of the bottom fraction.
• Method 2 (LCD Method): Find the Least Common Denominator (LCD) of all minor fractions within the expression. Multiply the main numerator and main denominator by this LCD to instantly clear all inner fractions.

To solve equations involving rational expressions, the most efficient method is often to clear the denominators by multiplying every term by the LCD.

This technique is incredibly useful in integral calculus, Laplace transforms, and control systems engineering. For a proper rational expression $\frac{P(x)}{Q(x)}$ (where the degree of $P(x)$ $\lt$ the degree of $Q(x)$), factor $Q(x)$ completely, then set up the partial fractions based on the factors:

• Distinct Linear Factors: For each factor $(ax + b)$, add a term $\frac{A}{ax + b}$.
• Repeated Linear Factors: For $(ax + b)^n$, add $\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \dots + \frac{A_n}{(ax + b)^n}$.
• Irreducible Quadratic Factors: For $(ax^2 + bx + c)$, add a term $\frac{Ax + B}{ax^2 + bx + c}$.

Rational inequalities take the form $\frac{P(x)}{Q(x)} \gt 0$, $\frac{P(x)}{Q(x)} \ge 0$, $\frac{P(x)}{Q(x)} \lt 0$, or $\frac{P(x)}{Q(x)} \le 0$. Similar to polynomial inequalities, we use a sign chart to analyze intervals.