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

How do you solve a rational equation?

Back

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

Front

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

Back

Checking for extraneous solutions is important because multiplying both sides of a rational equation by a variable expression can introduce solutions that do not satisfy the original equation.

Front

What is the first step in solving the equation \( \frac{n}{n-4}+n=\frac{12-4n}{n-4} \)?

Back

The first step is to find a common denominator, which in this case is \( n-4 \), and then multiply through by this denominator to eliminate the fractions.

Front

What are the solutions to the equation \( \frac{n}{n-4}+n=\frac{12-4n}{n-4} \)?

Back

The solutions are \( n=-4 \) and \( n=3 \).

Front

How do you isolate a variable in a rational equation?

Back

To isolate a variable in a rational equation, you can cross-multiply if the equation is in the form of two fractions set equal to each other, or you can manipulate the equation algebraically to get the variable on one side.

Front

What is the solution to the equation \( \frac{5}{x+2}=\frac{1}{x-4} \)?

Back

The solution is \( x=5.5 \).

Front

What is the method to solve \( \frac{5}{3}-\frac{2}{x}=\frac{8}{x} \)?

Back

To solve this equation, you can first combine the fractions on the left side and then isolate \( x \) by cross-multiplying.

Front

What is the solution to the equation \( \frac{5}{3}-\frac{2}{x}=\frac{8}{x} \)?

Back

The solution is \( x=6 \).

Front

What is the general form of a rational expression?

Back

A rational expression is generally in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.