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A student solved the following equation using the following steps:

4(2−3x)=x−2(2x+1)

8−3x=x−4x−2

8−3x=−3x−2

No solution.

Based on the student's work, the equation was solved__________________

The equation solved correctly would show that it has___________solution(s)

5 answers

Let's analyze the student's work step by step.

The original equation is: \[ 4(2 - 3x) = x - 2(2x + 1) \]

Now simplify each side:

  1. Left Side: \[ 4(2 - 3x) = 8 - 12x \]

  2. Right Side: \[ x - 2(2x + 1) = x - 4x - 2 = -3x - 2 \]

So the equation rewrites to: \[ 8 - 12x = -3x - 2 \]

Now, let's check the student's calculations:

  1. The next step taken by the student was: \[ 8 - 12x = -3x - 2 \] That matches our simplification.

  2. Combine like terms: Add \(12x\) to both sides: \[ 8 = 9x - 2 \]

  3. Then add 2 to both sides: \[ 10 = 9x \]

  4. Now, divide by 9: \[ x = \frac{10}{9} \]

The student's final claim was "No solution." However, we found: \[ x = \frac{10}{9} \]

This shows that the student made an error in their conclusion.

The equation was solved incorrectly.

The equation solved correctly would show that it has one solution (specifically, \( x = \frac{10}{9} \)).

Based on the student's work, the equation was solved was it incorrect or correct

Based on the student's work and the steps taken, the conclusion drawn by the student ("No solution") was incorrect.

The original equation was correctly simplified, and following through the algebra led to the solution \( x = \frac{10}{9} \). Therefore, the student's conclusion that there was "No solution" is incorrect.

Hence, the equation was solved incorrectly as the final conclusion does not reflect the correct solution found in the working steps.

The equation solved correctly would show that it has

two

one

three
imany

The equation solved correctly would show that it has one solution.

In summary, the correct solution to the equation \(4(2 - 3x) = x - 2(2x + 1)\) results in \(x = \frac{10}{9}\), indicating that there is exactly one solution.