Solve rational equations and respect domain restrictions. | Nonlinear equations in one variable | SAT Math

Rational equations need the LCD and a domain check before you accept any answer.

Core Idea

Multiply every term by the least common denominator (LCD) to clear fractions, solve the resulting equation, then exclude any solution that makes a denominator zero — those values were never in the domain.

Understanding

First check the domain. Any denominator that can become zero is off limits.

Identify the values that make each denominator zero.
Multiply both sides by the LCD to clear fractions.
Solve the resulting equation.
Throw out any answer that hits a restricted value.

Cross-multiplying works when you have $𝐴 𝐵 = 𝐶 𝐷$ , but the domain check still comes first.

Step by Step

Identify all denominators and note the values of $𝑥$ that make them zero (domain restrictions).
Multiply every term on both sides by the LCD to clear fractions.
Solve the resulting equation.
Discard any solution that equals a restricted value.

Misconceptions

Forgetting to check domain restrictions. A solution that makes a denominator zero is not valid, even if the algebra produces it.
Multiplying only one side of the equation by the LCD instead of both sides.
Not distributing the LCD to every term, especially when one side has addition or subtraction.

Question

Worked Example

What is the solution to $𝑥 𝑥 - 3 = 3 𝑥 - 3 + 2$ ?

Select an answer to see the explanation