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 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 rational 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 the solutions obtained from solving the equation do not make any denominator zero in the original equation.

Front

Solve the rational equation: \( \frac{2}{x} = \frac{4}{x+2} \).

Back

Cross-multiply to get \( 2(x+2) = 4x \). Simplifying gives \( 2x + 4 = 4x \), leading to \( x = 2 \).

Front

What is an extraneous solution?

Back

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

Front

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

Back

The least common denominator is the product of the unique factors of the denominators: \( x(x-1) \).

Front

What is the importance of simplifying rational expressions before solving?

Back

Simplifying rational expressions can make the equation easier to solve and helps to avoid mistakes in calculations.

Front

Solve the rational equation: \( \frac{x-1}{x+2} = \frac{3}{4} \).

Back

Cross-multiply to get \( 4(x-1) = 3(x+2) \). This simplifies to \( 4x - 4 = 3x + 6 \), leading to \( x = 10 \).