How many solutions does the equation 3x + 9 = 3x - 12 have?

Rationale

The equation 3x + 9 = 3x - 12 has 0 solutions.

When simplifying the equation, both sides have the term 3x, which cancels out. This results in the equation 9 = -12, a statement that is false. Therefore, there are no values for x that satisfy this equation.

A (0) CORRECT

This is the correct answer because, upon simplifying the equation to 9 = -12, we find a contradiction. Since no value of x can make this statement true, the equation has zero solutions.

B (1) YOUR ANSWER

This choice suggests that there is one unique solution to the equation. For an equation to have one solution, the terms would need to allow for a valid value of x. However, the simplification leads to a false statement, indicating that there is no solution at all.

C (2) YOUR ANSWER

Choosing 2 implies that the equation can yield two different values for x. However, with the simplification resulting in an impossible equality (9 = -12), it confirms that there are not even two solutions available.

D (3) YOUR ANSWER

This option indicates three solutions exist, which is incorrect. An equation can only have multiple solutions if it leads to a valid equality or a true statement. The contradiction found after simplification shows there are no solutions, not three.

E (Infinitely many) YOUR ANSWER

This answer would imply that there are countless solutions to the equation, which typically occurs in cases of identity (e.g., 0 = 0). However, since we arrived at a false statement, there are no solutions at all.

Conclusion

The equation 3x + 9 = 3x - 12 simplifies to an impossible statement, indicating that it has 0 solutions. The misleading nature of the choices highlights common misconceptions regarding the solution sets of linear equations. Understanding this principle clarifies how to approach similar equations in the future, reinforcing the importance of simplification in determining solvability.