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Justify why -12 (x-3) - 52 = -16 has one solution, infinitely many solutions, or no solutions.

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To analyze the equation \(-12 (x-3) - 52 = -16\), we begin by simplifying it step by step.

  1. Distribute the \(-12\) across \((x - 3)\): \[ -12(x - 3) = -12x + 36 \] Therefore, the equation becomes: \[ -12x + 36 - 52 = -16 \]

  2. Now combine the constants on the left side: \[ 36 - 52 = -16 \] So: \[ -12x - 16 = -16 \]

  3. Next, add \(16\) to both sides to isolate the term with \(x\): \[ -12x = -16 + 16 \] Which simplifies to: \[ -12x = 0 \]

  4. Finally, divide both sides by \(-12\): \[ x = 0 \]

Since we found a specific value for \(x\) (which is \(x = 0\)), we conclude that the equation has one solution.

In summary, the equation \(-12 (x-3) - 52 = -16\) has one solution, which is \(x = 0\).