To determine which of the steps could lead to the conclusion that the equation has no solution, we need to see if any of the steps results in a contradictory statement (such as a false statement like 0 = 1).
Let's examine each option:
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3(x + 4) = x + 4
Distributing on the left side gives: \(3x + 12 = x + 4\) Rearranging leads us to: \(3x - x = 4 - 12\) \(2x = -8\) \(x = -4\) (this has a solution, so this step does not indicate no solution)
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3(x + 4) = x + 12
Distributing on the left gives: \(3x + 12 = x + 12\) Rearranging leads us to: \(3x - x = 12 - 12\) \(2x = 0\) \(x = 0\) (this has a solution, so this step does not lead to no solution)
-
3(x + 4) = 3x + 12
Distributing on the left gives: \(3x + 12 = 3x + 12\) This simplifies to an identity, which is true for all values of \(x\). (indicates infinitely many solutions, thus does not indicate no solution)
-
3(x + 4) = 3x + 4
Distributing on the left gives: \(3x + 12 = 3x + 4\) Rearranging leads us to: \(3x - 3x = 4 - 12\) \(0 = -8\) (this is a contradiction, which means there is no solution)
Based on the analysis, the step that could indicate that the equation has no solution is:
3(x + 4) = 3x + 4