Under what situation would one or more solutions of a rational equation be unacceptable?
One or more solutions of a rational equation would be considered unacceptable when they create a contradiction or violate the conditions of the problem.
To understand this further, let's first clarify what a rational equation is. A rational equation is an equation that contains one or more rational expressions, where a rational expression is a fraction where the numerator and/or denominator is a polynomial. Rational equations may have one, none, or a finite number of solutions.
Now, here are three common situations in which the solutions of a rational equation can be considered unacceptable:
1. Division by zero: A solution that makes the denominator of a rational expression equal to zero is unacceptable. Dividing by zero is undefined and violates the basic rules of arithmetic. So, any solution that results in a zero in the denominator should be excluded.
2. Extraneous solutions: In some cases, a rational equation may yield solutions that are not valid solutions to the original problem. This happens because during the process of solving the equation, operations like squaring both sides or taking the square root are applied, which can introduce extraneous solutions. These solutions satisfy the equation mathematically but do not meet the conditions of the original problem. Therefore, they should be disregarded.
3. Contradictions: A solution that leads to a contradiction within the equation or the problem itself is also considered unacceptable. For example, if solving a rational equation results in an equation like "0 = 1" or "4 = 7", which are obviously false, then any solution obtained from that equation should be rejected.
In summary, one or more solutions of a rational equation may be unacceptable if they result in division by zero, are extraneous solutions that do not satisfy the original problem, or create contradictions within the equation or the problem. It is important to carefully analyze the solutions obtained to ensure they are valid and meaningful in the given context.