Solving Rational Equations

Front

What is a rational equation?

Back

A rational equation is an equation that involves at least one rational expression, which is a fraction where the numerator and denominator are polynomials.

Front

What are domain restrictions in rational equations?

Back

Domain restrictions are values of the variable that make the denominator zero, which are not allowed in rational equations.

Front

State the domain restrictions for the rational equation \( \frac{1}{x-2} \).

Back

x ≠ 2.

Front

How do you solve a rational equation?

Back

To solve a rational equation, find a common denominator, eliminate the fractions, and solve the resulting polynomial equation.

Front

What is the first step in solving \( \frac{2}{x+1} = \frac{3}{x-2} \)?

Back

Identify the domain restrictions: x ≠ -1 and x ≠ 2.

Front

What does it mean to cross-multiply in rational equations?

Back

Cross-multiplying means multiplying the numerator of one fraction by the denominator of the other fraction to eliminate the fractions.

Front

Solve the equation \( \frac{3}{x-1} = 5 \).

Back

x = 4.

Front

What is the importance of checking your solution in rational equations?

Back

Checking your solution ensures that it does not violate any domain restrictions and is a valid solution.

Front

What happens if you ignore domain restrictions when solving rational equations?

Back

Ignoring domain restrictions can lead to extraneous solutions that are not valid in the context of the original equation.

Front

Provide an example of a rational equation with no solution.

Back

\( \frac{1}{x} + 1 = \frac{1}{x} \) has no solution because it simplifies to 1 = 0.