Solve Rational Equation

Algebra-2

1. Fundamental Concepts

Definition: Rational equations are equations involving fractions with variables in the denominator.
Common Denominator: The process of finding a common denominator to solve rational equations.
Extraneous Solutions: Solutions that do not satisfy the original equation due to domain restrictions.

2. Key Concepts

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

Solving Strategy: Multiply both sides by the least common denominator (LCD) to eliminate fractions.

Verification: Check solutions in the original equation to ensure they do not make the denominator zero.

3. Examples

Example 1 (Basic)

Problem: Solve $$\frac{x}{x-2} + \frac{3}{x+2} = 4$$

Step-by-Step Solution:

Find the LCD: $$(x-2)(x+2)$$
Multiply each term by the LCD: $$x(x+2) + 3(x-2) = 4(x-2)(x+2)$$
Simplify and solve for \(x\): $$x^2 + 2x + 3x - 6 = 4(x^2 - 4)$$
Combine like terms: $$x^2 + 5x - 6 = 4x^2 - 16$$
Rearrange and solve: $$0 = 3x^2 - 5x - 10$$
Use the quadratic formula: $$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(-10)}}{2(3)}$$
Solutions: $$x = \frac{5 \pm \sqrt{25 + 120}}{6} = \frac{5 \pm \sqrt{145}}{6}$$

Validation: Substitute \(x = \frac{5 + \sqrt{145}}{6}\) into the original equation to check if it satisfies the equation.

Example 2 (Intermediate)

Problem: Solve $$\frac{2}{x-3} = \frac{5}{x+3} + \frac{1}{x^2-9}$$

Step-by-Step Solution:

Factor the denominators: $$x^2 - 9 = (x-3)(x+3)$$
Find the LCD: $$(x-3)(x+3)$$
Multiply each term by the LCD: $$2(x+3) = 5(x-3) + 1$$
Simplify and solve for \(x\): $$2x + 6 = 5x - 15 + 1$$
Rearrange and solve: $$21 = 3x$$
Solution: $$x = 7$$

Validation: Substitute \(x = 7\) into the original equation to check if it satisfies the equation.

4. Problem-Solving Techniques

Identify Common Denominators: Always start by identifying the least common denominator (LCD).
Eliminate Fractions: Multiply both sides of the equation by the LCD to clear fractions.
Check for Extraneous Solutions: Always verify solutions in the original equation to ensure they are valid.