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/or 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 find a common denominator, which is \((x-2)(x+1)\), and then multiply both sides by this common denominator.

Front

What does it mean to check for extraneous solutions?

Back

Checking for extraneous solutions means substituting the found solutions back into the original equation to ensure they do not make any denominator zero.

Front

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

Back

x = 7.

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

Solve for x: \( \frac{2}{x} + 3 = 5 \)

Back

x = 1.

Front

What is the importance of finding a common denominator in rational equations?

Back

Finding a common denominator allows you to eliminate fractions, making it easier to solve the equation.

Front

Solve for x: \( \frac{x-4}{x+2} = 0 \)

Back

x = 4.

Front

What is the definition of a polynomial?

Back

A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents.