Which of the following is false about non-reversible operations?
A A non-reversible operation is one that you can do in one direction, but the reverse is not always true
B Some examples of non-reversible operations include raising a variable to an even power and multiplying by a variable expression
C A non-reversible operation may lead to an extraneous solution, but not always
D After using a non-reversible operation to solve an equation, there will always be an extraneous solution
The correct answer and explanation is:
The correct answer is D: After using a non-reversible operation to solve an equation, there will always be an extraneous solution.
Explanation:
Non-reversible operations are mathematical steps that, when performed on an equation, can lead to results that aren’t always valid when the operation is undone. Common examples of non-reversible operations include squaring both sides of an equation or taking even roots. The reason these operations are non-reversible is that they may introduce solutions that are not valid in the context of the original equation.
Let’s break down the options:
- A: A non-reversible operation is one that you can do in one direction, but the reverse is not always true.
This statement is true. Non-reversible operations do not always allow you to go back to the original equation after performing them. For example, squaring both sides of an equation eliminates the sign of a number, so the reverse operation (taking the square root) does not always yield the same result as the original equation. - B: Some examples of non-reversible operations include raising a variable to an even power and multiplying by a variable expression.
This statement is true. Raising a variable to an even power (like squaring) and multiplying by a variable expression are both non-reversible. In both cases, solving the equation after applying these operations can give extraneous solutions or incorrect results if reversed. - C: A non-reversible operation may lead to an extraneous solution, but not always.
This statement is true. While non-reversible operations can lead to extraneous solutions, they do not always do so. For example, if you square both sides of an equation, the solutions you find may or may not be extraneous. You need to check each solution by substituting it back into the original equation to ensure it works. - D: After using a non-reversible operation to solve an equation, there will always be an extraneous solution.
This statement is false. While non-reversible operations can lead to extraneous solutions, it is not guaranteed. Not every application of a non-reversible operation will result in an extraneous solution. The need to check solutions depends on the type of operation and the specifics of the equation being solved.
In summary, non-reversible operations have the potential to introduce extraneous solutions, but this does not mean that every time they are used, extraneous solutions will appear. Therefore, option D is false.