Let's break down the steps and provide the necessary statements and reasons for solving the equation \(-2(x - 4) = 2x + 12\) to prove that \(x = -1\).
Proof Steps
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Statement 1: \[-2(x - 4) = 2x + 12\] Reason: Given.
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Statement 2: \[-2x + 8 = 2x + 12\] Reason: Distribute \(-2\) on the left side.
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Statement 3: \[-2x + 8 - 12 = 2x + 12 - 12\] Reason: Subtract \(12\) from both sides.
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Statement 4: \[-2x - 4 = 2x\] Reason: Simplify left side.
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Statement 5: \[-2x - 4 - 2x = 2x - 2x\] Reason: Subtract \(2x\) from both sides.
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Statement 6: \[-4 - 4x = 0\] Reason: Combine like terms on the left side.
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Statement 7: \[-4x = 4\] Reason: Add \(4\) to both sides.
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Statement 8: \(x = \frac{4}{-4}\) Reason: Divide both sides by \(-4\).
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Statement 9: \(x = -1\) Reason: Simplify \(\frac{4}{-4}\).
Conclusion
Thus, we have shown that \(x = -1\) through algebraic manipulation of the given equation.
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