Let's analyze the steps provided by the student to check the work and answer the questions.
- Check the student's steps:
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Starting equation: \( 4(2 - 3x) = x - 2(2x + 1) \)
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Expanding both sides: \[ 8 - 12x = x - 4x - 2 \] Simplifying the right side: \[ 8 - 12x = -3x - 2 \]
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Rearranging gives: \[ 8 - 12x = -3x - 2 \]
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Adding \( 12x \) and \( 3x \) to both sides: \[ 8 = 9x - 2 \]
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Adding 2 to both sides: \[ 10 = 9x \]
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Dividing by 9 gives: \[ x = \frac{10}{9} \]
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Thus, the equation has a unique solution, \( x = \frac{10}{9} \), not "No solution."
Answer to Question 3
The equation solved correctly would show that it has 1 solution.
Question 4
Equation with infinitely many solutions: An equation has infinitely many solutions if it simplifies to an identity, such as:
\[ 0 = 0 \]
For example: \[ 2x + 3 = 2(x + \frac{3}{2}) \]
This simplifies to the same expression on both sides.
To select from multiple-choice options (if provided), look for an equation where both sides can be manipulated to yield a tautology like \( 0 = 0 \).