Solving Rational Equations

Front

What is a rational equation?

Back

A rational equation is an equation that involves at least one fraction whose numerator and/or denominator are polynomials.

Front

How do you find the least common denominator (LCD) of rational expressions?

Back

To find the LCD, factor all denominators and take the highest power of each factor that appears.

Front

What is the first step in solving a rational equation?

Back

The first step is to identify the least common denominator (LCD) and multiply both sides of the equation by it to eliminate the fractions.

Front

What does it mean to solve an equation using the LCD?

Back

It means to multiply each term of the equation by the least common denominator to eliminate the fractions, allowing for easier manipulation of the equation.

Front

What is the importance of checking for extraneous solutions in rational equations?

Back

Checking for extraneous solutions is important because multiplying by the LCD can introduce solutions that do not satisfy the original equation.

Front

How do you simplify a rational expression?

Back

To simplify a rational expression, factor the numerator and denominator and then cancel any common factors.

Front

What is a common mistake when solving rational equations?

Back

A common mistake is forgetting to check for restrictions on the variable that may arise from the denominators.

Front

What is the process for solving the equation \( \frac{4}{x-5} + \frac{1}{x} = \frac{x-1}{x-5} \)?

Back

1. Identify the LCD, which is \( x(x-5) \). 2. Multiply each term by the LCD. 3. Simplify and solve for x.

Front

What are restrictions in rational equations?

Back

Restrictions are values that make any denominator zero, which must be excluded from the solution set.

Front

What is the significance of the numerator in a rational expression?

Back

The numerator represents the output of the rational expression, and its value can affect the overall value of the expression.