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A student solved the following equation using the following steps:

2(2x + 2) = 4(x + 4)
4x + 4 = 4x + 4
4 = 4
Infinitely many solutions
Based on the student's work, the equation was solved _________
The equation solved correctly would show that it has ________ solution(s).
Part 1
A. correctly
B. incorrectly
Part 2
A. two
B. no
C. one
D. three

5 answers

Part 1: A. correctly
Part 2: A. two

Explanation:

  1. The student's manipulation of the equation was correct, leading to the conclusion that both sides are equal (4 = 4), which indicates an identity.
  2. Since the equation simplifies to a true statement for all values of \( x \), it actually has infinitely many solutions, but among the provided options for Part 2, none indicate "infinitely many solutions." However, the closest correct choice would be that it has "two" distinct forms of solutions, meaning for all \( x \) it equals to an identity rather than just one specific value.

Given the options and context, it's a bit misleading but based on traditional classification of solutions, the result reflects an infinite form that is represented incorrectly in the choice set.

choose which answer is correct in part 1 and part 2

Based on the student's work:

Part 1: A. correctly
The student solved the equation correctly as it simplifies to a true statement (4 = 4).

Part 2: B. no
Although the equation shows that there are infinitely many solutions, the most fitting choice from the given options is "no" because it indicates there is not a single, unique solution.

So the correct answers are:

  • Part 1: A. correctly
  • Part 2: B. no
are you sure these answers are correct

Let me clarify the answers based on the work presented:

  1. The student reached a result of \(4 = 4\), which indicates that the equation is true for all values of \(x\), meaning there are infinitely many solutions.

Now reviewing the options again:

Part 1: A. correctly
The student's algebraic manipulation was correct, resulting in a valid statement.

Part 2:
The options given were:

  • A. two
  • B. no
  • C. one
  • D. three

Since the equation simplifies to \(4 = 4\), which is true for all \(x\), the correct answer should indeed reflect that there are infinitely many solutions. However, since none of the provided options directly state "infinitely many," it can be misleading. But among the offered choices, none correctly expresses the nature of infinite solutions.

The most appropriate answer to Part 2 would be that there are no unique solutions among the provided options (since it indicates not a single solution).

So, the answers are:

  • Part 1: A. correctly
  • Part 2: B. no

However, keeping in mind none of the choices directly address "infinitely many solutions," if you need to select strictly from the options, "no" indicates that there is not just one unique solution, which aligns with the infinite nature of solutions. Thank you for your patience and for double-checking!