Solve the problem without using a calculator. Which of the...

Solve the problem without using a calculator.
Which of the following are the solutions of the equation $x-\frac{3 x}{x+2}=\frac{6}{x+2} ?$
(A) $x=-2$ or $x=3$
(B) $x=-1$ or $x=3$
(C) Only $x=-2$
(D) Only $x=3$
(E) There are no solutions.

Key Concepts

Rational Equations

Rational equations involve ratios of polynomials and require careful treatment because the variable appears in the denominator. They often necessitate techniques to combine fractions or clear denominators to transform the equation into a polynomial form that can be solved using standard methods.

Domain Restrictions

When dealing with rational expressions, it is important to identify values that make any denominator zero. These restrictions must be considered because any solution making the denominator zero is not valid, regardless of the algebraic manipulations performed.

Clearing Denominators

Clearing denominators is a technique used to eliminate fractions by multiplying both sides of the equation by the least common denominator. This step converts the rational equation into a polynomial equation, making it easier to solve. However, it is crucial to keep track of any domain restrictions introduced by the original denominators.

Extraneous Solutions

Extraneous solutions are possible when solving rational equations, especially after clearing denominators. Once a solution is obtained from the transformed equation, it must be checked against the original equation's domain restrictions to ensure that it does not lead to undefined expressions.

Quadratic Equations

Often, after clearing denominators in a rational equation, the resulting equation is quadratic. Solving quadratic equations can be done by factoring, completing the square, or using the quadratic formula. Once the quadratic is solved, each potential solution must be verified within the context of the original equation.

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