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 the domain restrictions for rational equations?

Back

The domain restrictions for rational equations are the values of the variable that make the denominator equal to zero, as these values are not allowed.

Front

State the domain restrictions for the 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 denominators by multiplying both sides by it, and then solve the resulting equation.

Front

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

Back

Identify the common denominator, which is (x-2)(x+1).

Front

What does it mean to check for extraneous solutions?

Back

Checking for extraneous solutions means verifying that the solutions obtained do not make any denominator in the original equation equal to zero.

Front

Solve the rational equation: \( \frac{2}{x+3} = \frac{4}{x-1} \).

Back

Cross-multiply to get: 2(x-1) = 4(x+3). Solve to find x = -5.

Front

What is an extraneous solution?

Back

An extraneous solution is a solution that emerges from the process of solving an equation but does not satisfy the original equation.

Front

How do you find the least common denominator (LCD) for \( \frac{1}{x-2} + \frac{1}{x+3} \)?

Back

The LCD is (x-2)(x+3).

Front

What is the importance of simplifying rational expressions before solving?

Back

Simplifying rational expressions can make it easier to identify and eliminate common factors, leading to a simpler solution process.