Rational Equations: Definitions, Restrictions, and Solution Methods

Rational Equations

Definition and Key Concepts

A rational equation is an equation in which a variable appears in the denominator of a fraction. Rational equations are a central topic in College Algebra, especially in the study of polynomial and rational functions.

• Rational Equation: An equation involving one or more rational expressions, typically with variables in the denominators.

• Example: $\frac{1}{x} = 12$

• To solve a rational equation, we often convert it into a linear equation by eliminating denominators.

Restrictions in Rational Equations

When solving rational equations, it is crucial to identify restrictions—values of the variable that make any denominator zero. These values are excluded from the solution set.

• Restriction: Any value that makes a denominator zero is not allowed as a solution.

• Zero Denominator: Division by zero is undefined, so such values must be excluded.

• Example: In $\frac{1}{x} = 12$, $x = 0$ is a restriction.

General Steps for Solving Rational Equations

Solving rational equations involves a systematic approach to ensure all solutions are valid and restrictions are considered.

• Step 1: Determine restrictions by setting each denominator equal to zero and solving for the variable.

• Step 2: Multiply both sides of the equation by the least common denominator (LCD) to eliminate fractions.

• Step 3: Solve the resulting linear (or polynomial) equation.

• Step 4: Check all solutions against the restrictions. Discard any solution that makes a denominator zero.

Step	Description
1	Determine restrictions by setting denominators equal to zero
2	Multiply by LCD to eliminate fractions
3	Solve the resulting equation
4	Check solutions with restrictions

Examples

Let us consider some examples to illustrate the process of solving rational equations.

• Example 1: $\frac{x}{x-1} = \frac{7}{6}$

  • Restriction: $x \neq 1$

  • Multiply both sides by $6(x-1)$ to clear denominators.

  • Solve for $x$ and check if the solution is not the restriction.

• Example 2: $\frac{x-5}{x-2} = \frac{-3}{x-2} + 6$ and $x \neq 2$

  • If the solution equals the restriction, there is no solution.

Practice Problems

• Practice 1: $\frac{2x+4}{x-1} = 5$

• Practice 2: $\frac{5}{x} = \frac{2}{3x} + \frac{4}{3x}$

• Practice 3: $\frac{-5}{x+4} = \frac{3}{x+4} = 5$

Special Case: Solution Equal to Restriction

If the solution to the equation is equal to a restriction, then the equation has no solution. The solution set is the empty set: $\emptyset$.

• Always check your solutions against the restrictions before finalizing your answer.

Summary Table: Rational Equation Solution Process

Step	Action
1	Identify restrictions (denominator = 0)
2	Multiply by LCD
3	Solve the equation
4	Check solutions with restrictions

Additional info: Rational equations are a subset of polynomial and rational functions, and their solution methods are foundational for later topics such as systems of equations and advanced algebraic manipulation.