To determine the number of solutions for the equation \( 9(x+1)=3(5x+1)−12 \), we'll start by simplifying and solving for \( x \).
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Distribute on both sides: \[ 9x + 9 = 15x + 3 - 12 \] This simplifies to: \[ 9x + 9 = 15x - 9 \]
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Rearranging gives: \[ 9 + 9 = 15x - 9x \] So, \[ 18 = 6x \]
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Now, divide both sides by 6: \[ x = 3 \]
Now that we have found \( x = 3 \), we can determine the number of solutions. Since we found one specific value of \( x \), the equation has one solution.
Thus, the answer is: The equation has one solution: \( x = 3 \).