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

What are domain restrictions in rational equations?

Back

Domain restrictions are values of the variable that make the denominator zero, which are not allowed in rational equations.

Front

State the domain restrictions for the rational equation \( \frac{1}{x-2} \).

Back

x ≠ 2.

Front

How do you solve a rational equation?

Back

To solve a rational equation, find a common denominator, eliminate the denominators by multiplying both sides by it, and then solve the resulting equation.

Front

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

Back

Identify the domain restrictions: x ≠ 2 and x ≠ -1.

Front

What does it mean to cross-multiply in rational equations?

Back

Cross-multiplying means multiplying the numerator of one fraction by the denominator of the other fraction to eliminate the fractions.

Front

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

Back

Cross-multiply to get 2(x-1) = 4(x+3). Simplifying gives x = -5.

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 can you check if a solution is valid in a rational equation?

Back

Substitute the solution back into the original equation to see if it holds true and does not violate any domain restrictions.

Front

What is the least common denominator (LCD)?

Back

The least common denominator is the smallest expression that can be used as a common denominator for all fractions in an equation.