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 is a polynomial.

Front

What does LCD stand for in the context of rational equations?

Back

LCD stands for Least Common Denominator, which is the smallest multiple of the denominators of the fractions in an equation.

Front

How do you find the LCD of multiple fractions?

Back

To find the LCD, identify the unique factors in all denominators, take the highest power of each factor, and multiply them together.

Front

What is the first step in solving a rational equation?

Back

The first step is to identify the denominators and find the LCD of all the fractions involved.

Front

What happens when you multiply both sides of a rational equation by the LCD?

Back

Multiplying both sides by the LCD eliminates the fractions, simplifying the equation to a polynomial form.

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 solve the equation \( \frac{8}{x} + 3 = \frac{x+8}{x-4} \)?

Back

Multiply both sides by the LCD, which is \( x(x-4) \), then simplify and solve for x.

Front

What is the LCD for the equation \( \frac{2}{x} + \frac{1}{x^2+x} = \frac{x+2}{4x^2} \)?

Back

The LCD is \( 4x^2(x+1) \).

Front

What is the significance of the denominator in a rational equation?

Back

The denominator cannot be zero, as this would make the expression undefined.

Front

What is the process of simplifying a rational expression?

Back

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