Solve Rational Equation - Word Problem - 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: $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

Degree Preservation: The highest degree in the result matches input

Application: Used to solve real-world problems involving rates and proportions

3. Examples

Example 1 (Basic)

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

Step-by-Step Solution:

Find a common denominator: $$(x-2)(x+2)$$
Multiply each term by the common denominator: $$x(x+2) + 3(x-2) = 4(x^2 - 4)$$
Simplify and solve for $x$: $$x^2 + 2x + 3x - 6 = 4x^2 - 16$$
Rearrange terms: $$0 = 3x^2 - 5x - 10$$
Solve using the quadratic formula: $$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(-10)}}{2(3)}$$

Validation: Substitute $x = \text{{values}}$ into the original equation to check.

Example 2 (Intermediate)

Problem: A river flows at a speed of 4 miles per hour. A boat travels 12 miles upstream and then returns downstream in a total time of 3.5 hours. Find the speed of the boat in still water.

Step-by-Step Solution:

Let $v$ be the speed of the boat in still water. The speed upstream is $v-4$ and downstream is $v+4$.
The time taken upstream is $\frac{12}{v-4}$ and downstream is $\frac{12}{v+4}$.
Total time equation: $$\frac{12}{v-4} + \frac{12}{v+4} = 3.5$$
Solve for $v$: $$\frac{12(v+4) + 12(v-4)}{(v-4)(v+4)} = 3.5$$
Simplify and solve: $$24v = 3.5(v^2 - 16)$$

Validation: Substitute $v = \text{{values}}$ into the original equation to check.

4. Problem-Solving Techniques

Visual Strategy: Draw diagrams to represent the problem visually.
Error-Proofing: Always check solutions in the original equation to avoid extraneous solutions.
Concept Reinforcement: Practice with a variety of problems to reinforce understanding.