Before solving an equation, how can you recognize that there may be extraneous solutions?
A There is no way of knowing if an equation may lead to extraneous solutions before solving
B If there is an expression that requires a non-reversible operation to solve for x, such as a square root
C If you know you have to use an non-reversible operation as the first step
D If there is an expression that requires a non-reversible operation, whether or not x is within that expression’=
The correct answer and explanation is:
The correct answer is D: If there is an expression that requires a non-reversible operation, whether or not x is within that expression.
Explanation:
Extraneous solutions can sometimes appear in equations that involve operations that are not reversible, such as squaring both sides of an equation or taking the square root of both sides. These operations can introduce solutions that are not valid when substituted back into the original equation.
When solving an equation, you need to be mindful of the operations you perform. If you square both sides of an equation, for example, you may end up with a solution that satisfies the squared equation but not the original equation. This is because squaring is a non-reversible operation. For example, both 3 and -3 satisfy the equation x2=9x^2 = 9, but when you check them in the original equation, one may not work if there were restrictions, such as domain restrictions that only allowed positive values.
The key point here is that the equation itself may require an operation that alters the original problem in a way that introduces solutions that wouldn’t have existed in the context of the original equation. In this case, the operation is non-reversible because you cannot simply “undo” squaring or other similar operations without potentially losing information about the valid solutions.
Another example is when dealing with rational equations where denominators cannot be zero. If you manipulate the equation by multiplying both sides, you might introduce solutions that would make a denominator zero, which would not be valid.
Thus, it is important to carefully check all potential solutions by substituting them back into the original equation to confirm that they do not violate any conditions that might lead to extraneous solutions.