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

How do you solve a rational equation?

Back

To solve a rational equation, first find a common denominator, eliminate the fractions by multiplying both sides by that denominator, and then solve the resulting polynomial equation.

Front

What is a common denominator?

Back

A common denominator is a shared multiple of the denominators of two or more fractions, allowing them to be combined or compared.

Front

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

Back

The first step is to find a common denominator, which in this case is \( x(x+1) \).

Front

What does it mean to check for extraneous solutions?

Back

Checking for extraneous solutions means verifying that any solutions found in the process of solving the equation do not make the original equation undefined.

Front

What is an example of an extraneous solution?

Back

If solving \( \frac{1}{x} = 2 \) gives \( x = \frac{1}{2} \), substituting back shows it is valid, but if \( x = 0 \) is found, it is extraneous because it makes the original equation undefined.

Front

What is the importance of factoring in solving rational equations?

Back

Factoring helps simplify the equation, making it easier to find solutions by breaking down polynomials into their roots.

Front

How do you find the restrictions on the variable in a rational equation?

Back

Restrictions are found by setting the denominators equal to zero and solving for the variable, as these values cannot be included in the solution.

Front

What is the solution to the equation \( \frac{x-1}{x+2} = 3 \)?

Back

To solve, multiply both sides by \( x+2 \) to get \( x-1 = 3(x+2) \). Simplifying gives \( x = 5 \).

Front

What is a rational expression?

Back

A rational expression is a fraction where both the numerator and the denominator are polynomials.