Question

3. 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.(2 points)Based on the student's work, the equation was solved The equation solved correctly would show that it has  solution(s).

Answers

Answered by GPT-4o mini
Let's analyze the given equation and the student’s steps to see if the equation can be solved correctly and if a solution exists.

The original equation is:

\[
4(2 - 3x) = x - 2(2x + 1)
\]

Now, let's go through the student's steps:

1. Distributing on both sides:
- Left side: \(4(2 - 3x) = 8 - 12x\)
- Right side: \(x - 2(2x + 1) = x - 4x - 2 = -3x - 2\)

So, the equation becomes:

\[
8 - 12x = -3x - 2
\]

2. The student then rewrites it as:

\[
8 - 3x = -3x - 2
\]

This step appears to be incorrect. The correct step is to keep it as \(8 - 12x = -3x - 2\).

3. To isolate \(x\), we can add \(12x\) to both sides:

\[
8 = -3x + 12x - 2
\]

4. Simplifying this gives us:

\[
8 = 9x - 2
\]

5. Adding 2 to both sides:

\[
10 = 9x
\]

6. Dividing both sides by 9 gives:

\[
x = \frac{10}{9}
\]

Thus, the correct solution to the equation \(4(2-3x) = x - 2(2x+1)\) is \(x = \frac{10}{9}\).

The student's conclusion of "No solution" is incorrect. The equation does indeed have a solution, which is \(x = \frac{10}{9}\).

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