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

Back

The first step is to cross-multiply to eliminate the fractions: \( 1 \cdot (x+1) = 3 \cdot (x-2) \).

Front

What does it mean if a rational equation has no solution?

Back

A rational equation has no solution if the variable values that satisfy the equation make any denominator zero.

Front

What is a common mistake when solving rational equations?

Back

A common mistake is failing to check for extraneous solutions that may arise from multiplying both sides by a variable expression.

Front

Solve for x: \( \frac{x+1}{x-3} = 2 \)

Back

x = 5.

Front

What is an extraneous solution?

Back

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

Front

Solve for x: \( \frac{2}{x+4} = \frac{3}{x-1} \)

Back

x = -2.

Front

What is the importance of checking solutions in rational equations?

Back

Checking solutions ensures that they do not make any denominators zero, which would invalidate the solution.

Front

Solve for x: \( \frac{x-1}{x+2} = 3 \)

Back

x = 7.