To determine the number of solutions for the equation \( \frac{2}{3}(6m + 9) - 6 = 4m \), let's solve it step by step.
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Start with the equation:
\[ \frac{2}{3}(6m + 9) - 6 = 4m \]
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Distribute \( \frac{2}{3} \) to \( (6m + 9) \):
\[ \frac{2}{3} \cdot 6m + \frac{2}{3} \cdot 9 - 6 = 4m \]
This simplifies to:
\[ 4m + 6 - 6 = 4m \]
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The \( + 6 \) and \( - 6 \) cancel out:
\[ 4m = 4m \]
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This equation is always true for any value of \( m \). Therefore, it indicates that there are infinite solutions.
Hence, the answer is infinite/many solutions.