Add Rational Expressions - Algebra-2

1. Fundamental Concepts

Definition: Rational expressions are fractions where the numerator and denominator are polynomials.
Common Denominator: The least common multiple of the denominators is used to add or subtract rational expressions.
Simplification: After adding or subtracting, the resulting expression should be simplified by factoring and canceling out common factors.

2. Key Concepts

Basic Rule: $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

Degree Preservation: The degree of the polynomial in the numerator and denominator remains consistent with the original expressions after addition or subtraction.

Application: Used in various applications such as solving real-world problems involving rates and proportions.

3. Examples

Example 1 (Basic)

Problem: Simplify $$\frac{3x}{x+2} + \frac{2x}{x-1}$$

Step-by-Step Solution:

Find a common denominator: $$(x+2)(x-1)$$
Rewrite each fraction with the common denominator: $$\frac{3x(x-1)}{(x+2)(x-1)} + \frac{2x(x+2)}{(x+2)(x-1)}$$
Add the numerators: $$\frac{3x(x-1) + 2x(x+2)}{(x+2)(x-1)}$$
Simplify the numerator: $$\frac{3x^2 - 3x + 2x^2 + 4x}{(x+2)(x-1)} = \frac{5x^2 + x}{(x+2)(x-1)}$$

Validation: Substitute $x=3$ → Original: $\frac{9}{5} + \frac{6}{2} = \frac{9}{5} + 3 = \frac{24}{5}$; Simplified: $\frac{45 + 3}{5} = \frac{48}{5} = \frac{24}{5}$ ✓

Example 2 (Intermediate)

Problem: $$\frac{4y}{y^2 - 1} - \frac{3}{y+1}$$

Step-by-Step Solution:

Factor the denominator: $$y^2 - 1 = (y+1)(y-1)$$
Rewrite the second term with the common denominator: $$\frac{4y}{(y+1)(y-1)} - \frac{3(y-1)}{(y+1)(y-1)}$$
Subtract the numerators: $$\frac{4y - 3(y-1)}{(y+1)(y-1)}$$
Simplify the numerator: $$\frac{4y - 3y + 3}{(y+1)(y-1)} = \frac{y + 3}{(y+1)(y-1)}$$

Validation: Substitute $y=2$ → Original: $\frac{8}{3} - 1 = \frac{5}{3}$; Simplified: $\frac{2 + 3}{3} = \frac{5}{3}$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use color-coding for different terms and factors to keep track of like terms and common denominators.
Error-Proofing: Always check the final answer by substituting a value for the variable to ensure the solution is correct.
Concept Reinforcement: Practice with a variety of problems that involve different types of rational expressions to reinforce understanding.