Let's analyze the steps given in solving the equation:
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The original equation is: \( 3x - 9x + 1 = 2(-3x + 1) - 1 \).
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After simplifying both sides, you arrive at: \( -6x + 1 = -6x + 2 - 1 \) which simplifies to \( -6x + 1 = -6x + 1 \).
Since both sides of the equation are equal (i.e., \( -6x + 1 = -6x + 1 \)), this means the equation holds true for all values of \( x \). Therefore, there are infinitely many solutions to the equation since the equation is an identity.
The true statement is:
There are infinitely many solutions to the equation.