To find how many solutions the equation \( \frac{2 + 6x}{2} = 3x + 3 \) has, we need to solve it step by step.
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First, let's simplify the left side of the equation: \[ \frac{2 + 6x}{2} = \frac{2}{2} + \frac{6x}{2} = 1 + 3x \]
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Now, the equation can be rewritten as: \[ 1 + 3x = 3x + 3 \]
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Next, we will isolate the terms involving \(x\): \[ 1 + 3x - 3x = 3 \] \[ 1 = 3 \]
This statement \(1 = 3\) is a contradiction, which means that the original equation has no solutions.
Thus, the equation \( \frac{2 + 6x}{2} = 3x + 3 \) has no solutions.