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/or denominator are polynomials.

Front

What are the domain restrictions for the rational expression \( \frac{1}{x-2} \)?

Back

x ≠ 2, because the denominator cannot be zero.

Front

How do you find the least common denominator (LCD) of the rational expressions \( \frac{1}{x-2} \) and \( \frac{1}{x+1} \)?

Back

The LCD is (x-2)(x+1).

Front

What is the first step in solving a rational equation?

Back

Identify and state the domain restrictions to avoid division by zero.

Front

How do you solve the equation \( \frac{1}{x-2} = \frac{3}{x+1} \)?

Back

Cross-multiply to get: 1(x+1) = 3(x-2). Then solve for x.

Front

What does it mean to simplify a rational expression?

Back

To simplify a rational expression means to reduce it to its lowest terms by canceling common factors in the numerator and denominator.

Front

What is the result of multiplying two rational expressions \( \frac{a}{b} \times \frac{c}{d} \)?

Back

The result is \( \frac{ac}{bd} \).

Front

What is the process for adding rational expressions with different denominators?

Back

Find the least common denominator (LCD), rewrite each expression with the LCD, and then combine the numerators.

Front

What is a vertical asymptote in the context of rational functions?

Back

A vertical asymptote occurs at values of x that make the denominator zero, indicating where the function approaches infinity.

Front

How do you determine if a rational expression is undefined?

Back

A rational expression is undefined when its denominator equals zero.